Fuzzy Group Theory

John N. Mordeson, Kiran R. Bhutani, Azriel Rosenfeld Fuzzy Group Theory Studies in Fuzziness and Soft Computing, Volum...

Author: John N. Mordeson | Kiran R. Bhutani | A. Rosenfeld

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John N. Mordeson, Kiran R. Bhutani, Azriel Rosenfeld Fuzzy Group Theory

Studies in Fuzziness and Soft Computing, Volume 182 Editor-in-chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: springeronline.com Vol. 168. Yap P. Tan, Kim H. Yap, Lipo Wang (Eds.) Intelligent Multimedia Processing with Soft Computing, 2005 ISBN 3-540-22902-7 Vol. 169. C.R. Bector, Suresh Chandra Fuzzy Mathematical Programming and Fuzzy Matrix Games, 2005 ISBN 3-540-23729-1 Vol. 170. Martin Pelikan Hierarchical Bayesian Optimization Algorithm, 2005 ISBN 3-540-23774-7 Vol. 171. James J. Buckley Simulating Fuzzy Systems, 2005 ISBN 3-540-24116-7 Vol. 172. Patricia Melin, Oscar Castillo Hybrid Intelligent Systems for Pattern Recognition Using Soft Computing, 2005 ISBN 3-540-24121-3 Vol. 173. Bogdan Gabrys, Kauko Leiviskä, Jens Strackeljan (Eds.) Do Smart Adaptive Systems Exist?, 2005 ISBN 3-540-24077-2 Vol. 174. Mircea Negoita, Daniel Neagu, Vasile Palade Computational Intelligence: Engineering of Hybrid Systems, 2005 ISBN 3-540-23219-2

Vol. 175. Anna Maria Gil-Lafuente Fuzzy Logic in Financial Analysis, 2005 ISBN 3-540-23213-3 Vol. 176. Udo Seiffert, Lakhmi C. Jain, Patric Schweizer (Eds.) Bioinformatics Using Computational Intelligence Paradigms, 2005 ISBN 3-540-22901-9 Vol. 177. Lipo Wang (Ed.) Support Vector Machines: Theory and Applications, 2005 ISBN 3-540-24388-7 Vol. 178. Claude Ghaoui, Mitu Jain, Vivek Bannore, Lakhmi C. Jain (Eds.) Knowledge-Based Virtual Education, 2005 ISBN 3-540-25045-X Vol. 179. Mircea Negoita, Bernd Reusch (Eds.) Real World Applications of Computational Intelligence, 2005 ISBN 3-540-25006-9 Vol. 180. Wesley Chu, Tsau Young Lin (Eds.) Foundations and Advances in Data Mining, 2005 ISBN 3-540-25057-3 Vol. 181. Nadia Nedjah, Luiza de Macedo Mourelle Fuzzy Systems Engineering, 2005 ISBN 3-540-25322-X Vol. 182. John N. Mordeson, Kiran R. Bhutani, Azriel Rosenfeld Fuzzy Group Theory, 2005 ISBN 3-540-25072-7

John N. Mordeson Kiran R. Bhutani Azriel Rosenfeld

Fuzzy Group Theory

ABC

John N. Mordeson

Kiran R. Bhutani

Creighton University Center for Research in Fuzzy Mathematics and Computer Science Omaha, NE 68178 USA E-mail: [email protected]

Department of Mathematics The Catholic University of America 620 Michigan Avenue Washington DC 20064 USA E-mail: [email protected]

Azriel Rosenfeld † Center for Automation Research University of Maryland College Park, MD 20742 USA

Library of Congress Control Number: 2005925382

ISSN print edition: 1434-9922 ISSN electronic edition: 1860-0808 ISBN-10 3-540-25072-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-25072-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and TechBooks using a Springer LATEX macro package Cover design: E. Kirchner, Springer Heidelberg Printed on acid-free paper

SPIN: 10936443

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543210

Preface

Lotﬁ A. Zadeh introduced the notion of a fuzzy subset of a set in his paper published in 1965. Zadeh’s ideas marked a new direction and stirred the interest of researchers worldwide. It provided tools and an approach to model imprecision and uncertainty present in phenomena that do not have sharp boundaries. Rapid theoretical developments and practical applications based on the concept of a fuzzy subset were seen to emerge soon after that. In 1971, Azriel Rosenfeld used the notion of a fuzzy subset of a set to introduce the notion of a fuzzy subgroup of a group. Rosenfeld’s paper inspired the development of fuzzy abstract algebra. He also introduced fuzzy graphs, an area which has been growing actively since then. This is the ﬁrst book dedicated entirely to the rapidly growing ﬁeld of fuzzy group theory. It is not easy to present in a single 300 page book all that has been done in fuzzy group theory up-to-date. However, the authors have made a sincere eﬀort to present in a systematic way some results that have appeared in papers and conference proceedings (including some work by the authors themselves). We thank the researchers worldwide for their contributions to this growing ﬁeld and special thanks to those whose work is referenced in our book. We hope that the reader will ﬁnd this book crisp and not fuzzy in presentation, as well as rewarding and motivating for developing further results and applications of fuzzy group theory. The material presented in this book has been selected so as to make this a good reference for graduate students and researchers working in fuzzy group theory. The end of each chapter lists numerous references. While some of those have contributed to the material in the chapters, others are directly related to the material presented and so have been listed there. In Chapter 1, we ﬁrst present some basic material concerning fuzzy subsets of a set. We then introduce the notion of a fuzzy subgroup of a group and develop some concepts such as normal fuzzy subgroups and complete and weak direct products of fuzzy subgroups. We also present the notion of the fuzzy order of an element of a group.

VI

Preface

In Chapter 2, we present several fuzzy versions of Lagrange’s Theorem and Caley’s Theorem. We consider fuzzy quotient groups, characteristic fuzzy subgroups and conjugate fuzzy subgroups. The notion of the ascending central series of a fuzzy subgroup is presented in Chapter 3 and used to deﬁne nilpotency of a fuzzy subgroup. The notion of the descending central series of a fuzzy subgroup is also presented and used to deﬁne the nilpotency of a fuzzy subgroup. It is shown that these two deﬁnitions are not equivalent. The notion of commutators to generate the derived chain of a fuzzy subgroup is introduced and is used to deﬁne a solvable fuzzy subgroup. Fuzzy versions of well-known crisp results are presented in this chapter. Fuzzy subgroups of Hamiltonian, solvable, P -Hall, and nilpotent groups are examined in Chapter 4. The notions of generalized characteristic fuzzy subgroups, fully invariant fuzzy subgroups, and characteristic fuzzy subgroups are introduced. It is shown that if G is a ﬁnite group all of whose Sylow subgroups are cyclic, then a fuzzy subgroup of a group G is normal if and only if it is a generalized fuzzy subgroup of G. Normal fuzzy subgroups, quasinormal fuzzy subgroups, (p, q)-subgroups, fuzzy cosets, fuzzy conjugates and SL(p, q)-subgroups are also considered in this chapter. In Chapter 5, we present two approaches to show the existence of free fuzzy subgroups. One features the approach by Garzon and Muganda. In Chapter 6, we study fuzzy subgroups of Abelian groups. We develop the notions of independent generators, primary fuzzy subgroups, divisible fuzzy subgroups, and pure fuzzy subgroups. We determine a complete system of invariants for those fuzzy subgroups which are direct sums of fuzzy subgroups whose supports are cyclic. We also develop the notions of basic fuzzy subgroups and p-basic fuzzy subgroups. In Chapter 7, we introduce the notion of the fuzzy direct product of fuzzy subgroups deﬁned over subgroups of a group. These ideas are applied to the problem in group theory of obtaining conditions under which a group G can be expressed as the direct product of its normal subgroups. The number of fuzzy subgroups of certain ﬁnite Abelian groups with respect to a suitable equivalence relation as determined by Murali and Makamba are considered in Chapter 8. The Abelian groups under consideration are those which are direct sums of cyclic groups of prime order and those of order pn q m for distinct primes p and q and nonnegative integers n and m. In Chapter 9, we present the work of Tom Head concerning methods for deriving fuzzy theorems from crisp ones and embedding lattices of fuzzy subgroups into lattices of crisp subgroups. We also present the work of Ajmaal and Thomas concerning properties of lattices of fuzzy subgroups. The ﬁrst part of Chapter 10 is concerned with deriving membership functions from similarity relations. An algebraic approach for the construction of fuzzy subgroups is also considered. The chapter closes with some applications of fuzzy subgroups to a generalized recognition problem.

Preface

VII

We have done our best to provide the reader with a complete bibliography used for writing this book. We welcome comments and suggestions by the readers and apologize in advance if we inadvertently missed any source of reference in our bibliography. The authors are grateful to the staﬀs of Springer-Verlag, especially Frank Holzwarth, Gabriele Maas, Heather King, Janusz Kacprzyk and Dr. Thomas Ditzinger. We are indebted to Dr. Timothy Austin, Dean, Creighton College of Arts and Sciences and to Dr. and Mrs. George Haddix for their support of our work. We also wish to thank Professor Paul Wang of Duke University for his strong support of fuzzy mathematics. The ﬁrst author dedicates the book to his grandchildren, Emily and twins Emma and Jenna. The second author dedicates the book to her supportive husband Ravi, and loving sons Navin and Manoj. Together, we dedicate this book to the family of Professor Azriel Rosenfeld.

John Mordeson Creighton University Kiran R. Bhutani The Catholic University of America Azriel Rosenfeld University of Maryland

We were deeply saddened by the passing away of Professor Rosenfeld in February 2004. His presence would have added further strength to our book. We have lost a great collaborator, an outstanding scientist and a wonderful mentor. He will be missed sorely. John Mordeson Kiran R. Bhutani

Azriel Rosenfeld (1931-2004)

Azriel Rosenfeld was a tenured research Professor, a Distinguished University Professor (since 1995), and founder (1983) and Director of the Center for Automation Research, a department level unit of the College of Computer, Mathematical and Physical Sciences at the University of Maryland in College Park. He directed the Center until his retirement in June 2001. Upon his retirement he was designated a Distinguished University Professor Emeritus in the University’s Institute for Advanced Studies. He also held aﬃliate professorships in the Departments of Computer Science and Psychology and in the College of Engineering. He held a Ph. D. in mathematics from Columbia University (1957), rabbinic ordination (1952) and a Doctor of Hebrew Literature degree (1955) from Yeshiva University and honorary Doctor of Technology degrees from Linkoping University, Sweden (1980) and Oulu University, Finland (1994), and an honorary Doctor of Humane Letters degree from Yeshiva University (2000). Dr. Rosenfeld was widely regarded as the leading researcher in the world in the ﬁeld of computer image analysis. Over a period of more than 35 years he made many fundamental and pioneering contributions to nearly every area of that ﬁeld. He wrote the ﬁrst textbook in the ﬁeld (1969): was founding editor of its ﬁrst journal (1972); and was co-chairman of its ﬁrst international conference (1987). He published 30 books and over 600 book chapters and journal articles, and directed over 50 Ph. D. dissertations. He was a Fellow of the Institute of Electrical and Electronics Engineers (1971), and won its Emanuel Piore Award in 1985; he was a founding Fellow of the American Association for Artiﬁcial Intelligence (1990) and of the Association for Computing Machinery (1993); he was a Fellow of the Washington Academy of Sciences (1998), and won its Mathematics and Computer Science Award in 1988; he was a founding Director of the Machine Vision Association of the Society of Manufacturing Engineers (1985-8), won its President’s Award in 1987, and was a certiﬁed Manufacturing Engineer (1988); he was a founding member of the IEEE Computer Society’s Technical Committee

X

Azriel Rosenfeld (1931-2004)

on Pattern Analysis and Machine Intelligence (1965), served as its Chairman (1985-7), and received the Society’s Meritorious Service Award in 1986; and its Harry Goode Memorial Award in 1944 and became a Golden Core member of the Society in 1996; he received the IEEE Systems, Man, Cybernetics Norbert Wiener Award in 1995; he received an IEEE Standards Medallion in 1990, and the Electronic Imaging International Imager of the Year Award in 1991; he was a founding member of the Governing Board of the International Association for Pattern Recognition (1978-85), served as its President (1980-2), won its ﬁrst K. S. Fu Award in 1988, and became one of its founding Fellows in 1994; he was a Foreign Member of the Academy of Science of the German Democratic Republic (1988-92), and was a Corresponding Member of the National Academy of Engineering of Mexico (1982). In 1985, he served as chairman of a panel appointed by the National Research Council to brief the President’s Science Advisor on the subject of computer vision; he has also served (1985-8) as a member of the Vision Committee of the National Research Council. In 1982-3 and 1986-8 he served as a member of a task force appointed by the Defense Science Board to review the state of the art in automatic target recognition, and in 1993-4 he chaired a panel that conducted an assessment of foreign pattern recognition and image understanding research and development. We were deeply saddened by his death on Sunday, February 22, 2004. His love of knowledge, his passion and dedication to research, his generosity and kindness will always be remembered.

Contents

1

Fuzzy Subsets and Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fuzzy Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Normal Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Homomorphisms and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Complete and Weak Direct Products . . . . . . . . . . . . . . . . . . . . . . . 1.6 Fuzzy Order Relative to Fuzzy Subgroups . . . . . . . . . . . . . . . . . . 1.7 Fuzzy Orders in Cyclic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 9 15 20 31 36 37

2

Fuzzy Caley’s Theorem and Fuzzy Lagrange’s Theorem . . . . 2.1 Properties of Normal Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . 2.2 Characteristic Fuzzy Subgroups and Abelian Fuzzy Subgroups 2.3 Fuzzy Caley’s Theorem and Fuzzy Lagrange’s Theorem . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 45 52 60

3

Nilpotent, Commutator, and Solvable Fuzzy Subgroups . . . . 3.1 Commutative Fuzzy Subsets and Nilpotent Fuzzy Subgroups . . 3.2 Nilpotent Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Solvable Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 72 83 88

4

Characterization of Certain Groups and Fuzzy Subgroups . 91 4.1 Fuzzy Subgroups of Hamiltonian, Solvable, P -Hall, and Nilpotent Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 Characterization of Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . 99 4.3 Quasi-normal and Normal Fuzzy Subgroups . . . . . . . . . . . . . . . . . 104 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

XII

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5

Free Fuzzy Subgroups and Fuzzy Subgroup Presentations . . 119 5.1 Free Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.2 Presentations of Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.3 Constructing Free Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . 127 5.4 Free (s,t]-Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6

Fuzzy Subgroups of Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . 139 6.1 Minimal Generating Sets and Direct Sums . . . . . . . . . . . . . . . . . . 139 6.2 Independent Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.3 Primary Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.4 Divisible and Pure Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . 147 6.5 Invariants of Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.6 Basic and p-Basic Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . 158 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7

Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.1 Fuzzy Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7.2 Fuzzy p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.3 Fuzzy Subgroups Having Property ∗ . . . . . . . . . . . . . . . . . . . . . . . 183 7.4 Cyclic Fuzzy Subgroups and Cyclic Fuzzy p-subgroups . . . . . . . 186 7.5 Fuzzy p∗ -subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

8

Equivalence of Fuzzy Subgroups of Finite Abelian Groups . 201 8.1 A Relation on the Set of Fuzzy Subsets of a Set . . . . . . . . . . . . . 201 8.2 Fuzzy Subgroups of p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.3 Pad Keychains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 8.4 Zpn ⊕ Zqm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8.5 Sums, Unions, and Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 8.6 Fuzzy Subgroups of Inﬁnite Cyclic Groups . . . . . . . . . . . . . . . . . . 235 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

9

Lattices of Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 9.1 Embedding of Fuzzy Power Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 9.2 Representation of the Fuzzy Power Algebra . . . . . . . . . . . . . . . . . 241 9.3 The Metatheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 9.4 Uniﬁcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 9.5 Lattices of Fuzzy Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 9.6 The Metatheorem Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 9.7 Fuzzy Subgroups With The Sup Property . . . . . . . . . . . . . . . . . . 257 9.8 Lattices of Fuzzy Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Contents

XIII

10 Membership Functions From Similarity Relations . . . . . . . . . . 267 10.1 Similarity Relations and Membership Functions . . . . . . . . . . . . . 268 10.2 Level Subgroups, Cosets, and Equivalence Classes . . . . . . . . . . . 272 10.3 Representation of Membership Functions . . . . . . . . . . . . . . . . . . . 276 10.4 Fuzzy Subgroups Based on Group Properties . . . . . . . . . . . . . . . . 280 10.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

1 Fuzzy Subsets and Fuzzy Subgroups

The pioneering work of Zadeh on fuzzy subsets of a set in [53] and Rosenfeld on fuzzy subgroups of a group in [43] led to the fuzziﬁcation of algebraic structures. In this chapter, we begin the study of fuzzy subgroups of a group. We write ∧ for minimum or inﬁmum and ∨ for maximum or supremum. Throughout this book, we assume that the least upper bound of the empty set is 0 and the greatest lower bound of the empty set is 1. We let X, Y, and Z denote nonempty sets. We let N denote the set of positive integers, Z the set of all integers, Q the set of rational numbers, R the set of real numbers, and ∅ the empty set. Unless otherwise stated I denotes an arbitrary nonempty index set. Many of the results of Sections 1.1-1.5 are taken from Yu, Mordeson and Cheng [52]. An extensive list of references of the material used in the development of [52] can be found there, for example, [13], [30], [48] and [51]. Many related papers can be found in the references.

1.1 Fuzzy Subsets In this section, we present some basic concepts of fuzzy set theory. In particular, we present the important principle called the extension principle. Deﬁnition 1.1.1. A fuzzy subset of X is a function from X into [0, 1]. The set of all fuzzy subsets of X is called the fuzzy power set of X and is denoted by FP(X). Deﬁnition 1.1.2. Let µ ∈ FP(X). Then the set {µ(x) | x ∈ X} is called the image of µ and is denoted by µ(X) or Im(µ). The set {x | x ∈ X, µ(x) > 0}, is called the support of µ and is denoted by µ∗ . In particular, µ is called a ﬁnite fuzzy subset if µ∗ is a ﬁnite set, and an inﬁnite fuzzy subset otherwise. If µ ∈ FP(X), then µ is said to have the sup property if every subset of µ(X) has a maximal element. John Mordeson, Kiran R. Bhutani, Azriel Rosenfeld: Fuzzy Group Theory, StudFuzz 182, 1–39 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

2

1 Fuzzy Subsets and Fuzzy Subgroups

Deﬁnition 1.1.3. Let Y ⊆ X and a ∈ [0, 1]. We deﬁne aY ∈ FP(X) as follows: a for x ∈ Y aY (x) = 0 for x ∈ X\Y. In particular, if Y is a singleton, say {y}, then a{y} is called a fuzzy point (or fuzzy singleton), and is sometimes denoted by ya . Let 1Y denote the characteristic function of Y. If S is a set of fuzzy singletons, then we let foot(S) = {y ∈ X | ya ∈ S}. Deﬁnition 1.1.4. Let µ, ν ∈ FP(X). If µ(x) ≤ ν(x) ∀ x ∈ X, then µ is said to be contained in ν (or ν contains µ), and we write µ ⊆ ν (or ν ⊇ µ). If µ ⊆ ν and µ = ν, then µ is said to be properly contained in ν (or ν properly contains µ) and we write µ ⊂ ν (or ν ⊃ µ). Clearly, the inclusion relation ⊆ is a partial ordering on FP(X). Deﬁnition 1.1.5. Let µ, ν ∈ FP(X). Deﬁne µ ∪ ν and µ ∩ ν ∈ FP(X) as follows: ∀ x ∈ X, (µ ∪ ν)(x) = µ(x) ∨ ν(x), (µ ∩ ν)(x) = µ(x) ∧ ν(x). Then µ ∪ ν and µ ∩ ν are called the union and intersection of µ and ν, respectively. For any collection, {µi | i ∈ I}, of fuzzy subsets of X, where I is a nonempty index set, the least upper bound ∪i∈I µi and the greatest lower bound ∩i∈I µi of the µi ’s are given by ∀x ∈ X, (∪i∈I µi )(x) = ∨i∈I µi (x) (∩i∈I µi )(x) = ∧i∈I µi (x), respectively. We write ∩i∈I µi = ∩ni=1 µi = µ1 ∩ µ2 ∩ ... ∩ µn and ∪i∈I µi = ∪ni=1 µi = µ1 ∪ µ2 ∪ ... ∪ µn if I = {1, 2, ..., n}. Deﬁnition 1.1.6. Let µ ∈ FP(X). For a ∈ [0, 1], deﬁne µa as follows: µa = {x | x ∈ X, µ(x) ≥ a}. µa is called the a-cut (or a-level set ) of µ. It follows easily that for all µ, ν ∈ FP(X), (1) µ ⊆ ν, a ∈ [0, 1] ⇒ µa ⊆ νa , (2) a ≤ b, a, b ∈ [0, 1] ⇒ µb ⊆ µa , (3) µ = ν ⇔ µa = νa ∀a ∈ [0, 1]. The next two theorems state some basic properties of cuts. Their proofs are straightforward.

1.1 Fuzzy Subsets

3

Theorem 1.1.7. Suppose that {µi | i ∈ I} ⊆ FP(X). Then for any a ∈ [0, 1], (1) ∪i∈I (µi )a ⊆ (∪i∈I µi )a , (2) ∩i∈I (µi )a = (∩i∈I µi )a , Moreover, if I is ﬁnite, then we have equality in (1). Theorem 1.1.8. Let µ ∈ FP(X) and {ai | i ∈ I} be a non-empty subset of [0, 1]. Let b = ∧i∈I ai and c = ∨i∈I ai . Then the following assertions hold: (1) ∪i∈I µai ⊆ µb , (2) ∩i∈I µai = µc . Theorem 1.1.9. Let µ ∈ FP(X). Then µ = ∪a∈[0,1] aµa = ∪a∈µ(X) aµa . Proof. Let x ∈ X. Then (∪a∈[0,1] aµa )(x) = ∨a∈[0,1] aµa (x) = ∨{a ∈ [0, 1] | a ≤ µ(x)} = µ(x). Thus µ = ∪a∈[0,1] aµa . Similarly, µ = ∪a∈µ(X) aµa . Deﬁnition 1.1.10. Let I be a nonempty index set and let {Xi | i ∈ I}be a collection of nonempty sets. Let X denote the Cartesian product of the Xi ’s, namely, X= Xi = {(xi )i∈I | xi ∈ Xi , i ∈ I}. i∈I

Let µi ∈ FP(Xi ) for all i ∈ I. Deﬁne the fuzzy subset µ of X by µ(x) = ∧i∈I µi (xi ) ∀x = (xi )i∈I ∈ X. Then µ is called the complete direct product of the µi ’s and is denoted by ∼ µ= µi . i∈I

If I = {1, 2, . . . , n}, then Xi X= i∈I

= X1 × X2 × . . . × Xn = {(x1 , x2 , . . . , xn ) | xi ∈ Xi , i = 1, 2, . . . , n}, and we write

∼

2⊗ . . . ⊗µ n. µi = µ1 ⊗µ

i∈I

Clearly, if µi , νi ∈ FP(Xi ) with µi ⊆ νi for all i ∈ I, then ∼ i∈I

µi ⊆

∼ i∈I

νi .

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1 Fuzzy Subsets and Fuzzy Subgroups

Deﬁnition 1.1.11 (Extension Principle). Let f be a function from X into Y, and let µ ∈ FP(X) and ν ∈ FP(Y ). Deﬁne the fuzzy subsets f (µ) ∈ FP(Y ) and f −1 (ν) ∈ FP(X) by ∀y ∈ Y, ∨{µ(x) | x ∈ X, f (x) = y} if f −1 (y) = ∅, f (µ)(y) = 0 otherwise and ∀x ∈ X,

f −1 (ν)(x) = ν(f (x)).

Then f (µ) is called the image of µ under f and f −1 (ν) is called the preimage (or inverse image) of ν under f. Recall that in Deﬁnition 1.1.11, the least upper bound of the empty set is the zero element. Theorem 1.1.12. Let f be a function from X into Y and g a function from Y into Z. Then the following assertions hold. (1) For all µi ∈ FP(X), i ∈ I, f (∪i∈I µi ) = ∪i∈I f (µi ) and so µ1 ⊆ µ2 ⇒ f (µ1 ) ⊆ f (µ2 ) ∀µ1 , µ2 ∈ FP(X). (2) For all νj ∈ FP(Y ), j ∈ J, where J is a nonempty index set, f −1 (∪j∈J νj ) = ∪j∈J f −1 (νj ), f −1 (∩j∈J νj ) = ∩j∈J f −1 (νj ), and therefore ν1 ⊆ ν2 ⇒ f −1 (ν1 ) ⊆ f −1 (ν2 ) ∀ν1 , ν2 ∈ FP(Y ). (3) f −1 (f (µ)) ⊇ µ ∀µ ∈ FP(X). In particular, if f is an injection, then f −1 (f (µ)) = µ ∀µ ∈ FP(X). This means µ → f (µ) is an injection from FP(X) into FP(Y ) and ν → f −1 (ν) is a surjection from FP(Y ) onto FP(X). (4) f (f −1 (ν)) ⊆ ν ∀ν ∈ FP(Y ). In particular, if f is a surjection, then f (f −1 (ν)) = ν ∀ν ∈ FP(Y ), and therefore µ → f (µ) is a surjection from FP(X) onto FP(Y ) and ν → f −1 (ν) is an injection from FP(Y ) into FP(X). (5) f (µ) ⊆ ν ⇔ µ ⊆ f −1 (ν) ∀µ ∈ FP(X) and ∀ν ∈ FP(Y ). (6) g(f (µ)) = (g ◦ f )(µ) ∀µ ∈ FP(X) and f −1 (g −1 (ξ)) = (g ◦ f )−1 (ξ) ∀ξ ∈ FP(Z).

1.2 Fuzzy Subgroups

5

Proof. Using the extension principle Deﬁnition 1.1.11 and the deﬁnitions following Deﬁnition 1.1.5, one can show that (1) and (2) hold. To prove (3), consider any µ ∈ FP(X). Then f −1 (f (µ))(x) = f (µ)(f (x)) = ∨{µ(x ) | x ∈ X, f (x ) = f (x)} ⊇ µ(x) ∀x ∈ X. In particular, if f is an injection, then f −1 (f (µ))(x) = ∨{µ(x ) | x ∈ X, f (x ) = f (x)} = µ(x) ∀ x ∈ X. Hence (3) is true. To prove (4), we consider any ν ∈ FP(Y ). Then f (f −1 (ν))(y) = ∨{f −1 (ν)(x) | x ∈ X, f (x) = y} = ∨{ν(f (x)) | x ∈ X, f (x) = y} ν(y) for y ∈ f (X) = 0 otherwise ≤ ν(y) ∀ y ∈ Y. Thus f (f −1 (ν))(y) = ν(y) ∀ y ∈ Y if f is a surjection. Hence assertion (4) holds. Assertion (5) is an immediate consequence of assertions (1) through (4). To prove assertion (6), we consider any µ ∈ FP(X) and any z ∈ Z. Then g(f (µ))(z) = ∨{f (µ)(y) | y ∈ Y, g(y) = z} = ∨{∨{µ(x) | x ∈ X, f (x) = y} | y ∈ Y, g(y) = z} = ∨{µ(x) | x ∈ X, (g ◦ f )(x) = z} = (g ◦ f )(µ)(z) ∀ z ∈ Z. Further, for all ξ ∈ FP(Z) and ∀ x ∈ X, ((g ◦ f )−1 (ξ))(x) = ξ(g(f (x))) = g −1 (ξ)(f (x)) = f −1 (g −1 (ξ))(x).

1.2 Fuzzy Subgroups For the remainder of this chapter, G denotes an arbitrary group with a multiplicative binary operation and identity e. In order to deﬁne the notion of a fuzzy subgroup and to examine its properties, we introduce some operations on a fuzzy subset of a group G in terms of the group operation.

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1 Fuzzy Subsets and Fuzzy Subgroups

Deﬁnition 1.2.1. We deﬁne the binary operation “◦” on FP(G) and the unary operation −1 on FP(G) as follows: ∀µ, ν ∈ FP(G) and ∀x ∈ G, (µ ◦ ν)(x) = ∨{µ(y) ∧ ν(z) | y, z ∈ G, yz = x} and µ−1 (x) = µ(x−1 ). We call µ ◦ ν the product of µ and ν, and µ−1 the inverse of µ. It is easy to verify that the binary operation ◦ in Deﬁnition 1.2.1 is associative. The next theorem can be proved easily using previously deﬁned notions and thus we omit its proof. Theorem 1.2.2. Let µ, ν, µi ∈ FP(G), i ∈ I. Let a = ∨{µ(x) | x ∈ G}. Then the following assertions hold: (1) (µ ◦ ν)(x) = ∨y∈G (µ(y) ∧ ν(y −1 x)) = ∨y∈G (µ(xy −1 ) ∧ v(y)) ∀ x ∈ G; (2) (ay ◦ µ)(x) = µ(y −1 x) ∀x, y ∈ G; (3) (µ ◦ ay )(x) = µ(xy −1 ) ∀x, y ∈ G; (4) (µ−1 )−1 = µ; (5) µ ⊆ µ−1 ⇔ µ−1 ⊆ µ ⇔ µ = µ−1 ⇔ µ(x) ≤ µ(x−1 ) ∀x ∈ G ⇔ µ(x−1 ) ≤ µ(x) ∀x ∈ G ⇔ µ(x) = µ(x−1 ) ∀x ∈ G; (6) (7) (8) (9)

µ ⊆ ν ⇔ µ−1 ⊆ ν −1 ; (∪i∈I µi )−1 = ∪i∈I µ−1 i ; (∩i∈I µi )−1 = ∩i∈I µ−1 i ; (µ ◦ ν)−1 = ν −1 ◦ µ−1 .

Deﬁnition 1.2.3. Let µ ∈ FP(G). Then µ is called a fuzzy subgroup of G if (1) µ(xy) ≥ µ(x) ∧ µ(y) ∀ x, y ∈ G and (2) µ(x−1 ) ≥ µ(x) ∀ x ∈ G. Denote by F(G), the set of all fuzzy subgroups of G. If µ ∈ F(G), we let µ∗ = {x ∈ G | µ(x) = µ(e)} and recall from Deﬁnition 1.1.2 that µ∗ denotes the support of µ. If µ ∈ FP(G) satisﬁes condition (1) of Deﬁnition 1.2.3, then µ(xn ) ≥ µ(x) ∀x ∈ G, where n ∈ N. Also, µ satisﬁes conditions (1) and (2) of Deﬁnition 1.2.3 if and only if µ(xy −1 ) ≥ µ(x) ∧ µ(y) ∀x, y ∈ G.

1.2 Fuzzy Subgroups

7

Comment 1.2.4. If µ ∈ F(G) and H is a subgroup of G, then µ|H ∈ F(H). Lemma 1.2.5. Let µ ∈ F(G). Then ∀ x ∈ G, (1) µ(e) ≥ µ(x); (2) µ(x) = µ(x−1 ). Proof. Let x ∈ G. (1) µ(e) = µ(xx−1 ) ≥ µ(x) ∧ µ(x−1 ) ≥ µ(x) ∧ µ(x) = µ(x). (2) µ(x) = µ((x−1 )−1 ) ≥ µ(x−1 ) ≥ µ(x). Hence µ(x) = µ(x−1 ).

We note that if µ is a fuzzy subgroup of a group G and if x, y ∈ G with µ(x) = µ(y), then µ(xy) = µ(x) ∧ µ(y): Suppose µ(x) > µ(y). Then µ(y) = µ(x−1 xy) ≥ µ(x−1 ) ∧ µ(xy) = µ(x) ∧ µ(xy). Thus µ(y) ≥ µ(x) ∧ µ(xy) and since µ(x) ≥ µ(y), it follows that µ(y) ≥ µ(xy) ≥ µ(x)∧µ(y) = µ(y). Thus µ(xy) = µ(x)∧µ(y). A similar argument can be used for the case µ(y) > µ(x). Lemma 1.2.6. Let µ ∈ FP(G). Then µ is a fuzzy subgroup of G if and only if µa is a subgroup of G ∀ a ∈ µ(G) ∪ {b ∈ [0, 1] | b ≤ µ(e)}. Proof. Suppose µ is a fuzzy subgroup of G and let a ∈ µ(G). Since µ(e) ≥ µ(x) ∀ x ∈ G, e ∈ µa . Thus µa = ∅. Let x, y ∈ µa . Then µ(x) ≥ a and µ(y) ≥ a. Since µ is a fuzzy subgroup, µ(xy −1 ) ≥ µ(x) ∧ µ(y) ≥ a ∧ a = a. Hence xy −1 ∈ µa and so µa is a subgroup of G . Similarly, if a ≤ µ(e), then one can show that µa is a subgroup of G. Conversely, suppose µa is a subgroup of G ∀ a ∈ µ(G) ∪ {b ∈ [0, 1] | b ≤ µ(e)}. Then ∀a ∈ µ(G) we must have e ∈ µa and so it follows that µ(e) ≥ a. Let x, y ∈ G and let µ(x) = a and µ(y) = b. Let c = a ∧ b. Then x, y ∈ µc and c ≤ µ(e). By hypothesis, µc is a subgroup of G and so xy −1 ∈ µc . Hence µ(xy −1 ) ≥ c = a ∧ b = µ(x) ∧ µ(y). Thus µ is a fuzzy subgroup of G. The next three results can be proved using previous notions. Corollary 1.2.7. If µ ∈ F(G), then µ∗ is a subgroup of G. Theorem 1.2.8. If µ ∈ F(G), then µ∗ is a subgroup of G. Theorem 1.2.9. Let µ ∈ FP(G). Then µ ∈ F(G) if and only if µ satisﬁes the following conditions: (1) µ ◦ µ ⊆ µ; (2) µ−1 ⊆ µ (or µ−1 ⊇ µ, or µ−1 = µ). Theorem 1.2.10. Let µ, ν ∈ F(G). Then µ ◦ ν ∈ F(G) if and only if µ ◦ ν = ν ◦ µ. Proof. Suppose µ ◦ ν ∈ F(G). Then µ ◦ ν = µ−1 ◦ ν −1 = (ν ◦ µ)−1 = ν ◦ µ. Conversely, suppose that µ ◦ ν = ν ◦ µ. Then (µ ◦ ν)−1 = (ν ◦ µ)−1 = −1 µ ◦ ν −1 = µ ◦ ν and

8

1 Fuzzy Subsets and Fuzzy Subgroups

(µ ◦ ν) ◦ (µ ◦ ν) = µ ◦ (ν ◦ µ) ◦ ν = µ ◦ (µ ◦ ν) ◦ ν = (µ ◦ µ) ◦ (ν ◦ ν) ⊆ µ ◦ ν. Consequently, by Theorem 1.2.9, µ ◦ ν ∈ F(G).

Theorem 1.2.11. Let µ ∈ F(G) and H be a group. Suppose that f is a homomorphism of G into H. Then f (µ) ∈ F(H). Proof. Let u, v ∈ H. Suppose either u ∈ / f (G) or v ∈ / f (G). Then f (µ)(u) ∧ f (µ)(v) = 0 ≤ f (µ)(uv). Now assume u ∈ / f (G). Then u−1 ∈ / f (G). Thus f (µ)(u) = 0 = f (µ)(u−1 ). Now suppose u = f (x) and v = f (y) for some x, y ∈ G. Then (f (µ))(uv) = ∨{µ(z) | z ∈ G, f (z) = uv} ≥ ∨{µ(xy) | x, y ∈ G, f (x) = u, f (y) = v} ≥ ∨{µ(x) ∧ µ(y) | x, y ∈ G, f (x) = u, f (y) = v} = (∨{µ(x) | x ∈ G, f (x) = u}) ∧ (∨{µ(y) | y ∈ G, f (y) = v}) = (f (µ))(u) ∧ (f (µ))(u). Also (f (µ))(u−1 ) = ∨{µ(z) |z ∈ H, f (z) = u−1 } = ∨{µ(z −1 ) | z ∈ H, f (z −1 ) = u} = (f (µ))(u). Hence f (µ) ∈ F(H). Theorem 1.2.12. Let H be a group and ν ∈ F(H). Let f be a homomorphism of G into H. Then f −1 (ν) ∈ F(G). Proof. Let x, y ∈ G. Then f −1 (ν)(xy) = ν(f (xy)) = ν(f (x)f (y)) ≥ ν(f (x)) ∧ ν(f (y)) = f −1 (ν)(x)∧ f −1 (ν)(y). Further, f −1 (ν)(x−1 ) = ν(f (x−1 )) = ν(f (x)−1 ) = ν(f (x)) = f −1 (ν)(x). Hence f −1 (ν) ∈ F(G). Theorem 1.2.13. Let {µi | i ∈ I} ⊆ F(G). Then ∩i∈I µi ∈ F(G). Proof. Let x, y ∈ G. Then (∩i∈I µi )(xy −1 ) = ∧{µi (xy −1 )|i ∈ I} ≥ {µi (x) ∧ µi (y)|i ∈ I} = (∧{µi (x)|i ∈ I}) ∧ (∧{µi (y)|i ∈ I}) = (∩i∈I µi )(x) ∧ (∩i∈I µi )(y).

Deﬁnition 1.2.14. Let µ ∈ FP(G). Let µ = ∩{ν | µ ⊆ ν, ν ∈ F(G)}. Then µ is called the fuzzy subgroup of G generated by µ. Clearly, µ is the smallest fuzzy subgroup of G which contains µ. We now present another procedure of constructing µ . Deﬁne µ1 = µ and µn = µn−1 ◦ µ ∀n ∈ N, n > 1.

1.3 Normal Fuzzy Subgroups

9

Theorem 1.2.15. Let µ ∈ FP(G) and let a = ∧{η(e) | µ ⊆ η, η ∈ F(G)}. Then −1 n −1 n ) ) = ∪∞ ) . µ = ea ∪ (∪∞ n=1 (µ ∪ µ n=1 (ea ∪ µ ∪ µ ∞ Proof. Let ν = ea ∪ ∪n=1 (µ∪ µ−1 )n . −1 n For all x ∈ G, ν(x−1 ) = ea ∪ (∪∞ ) (x−1 ) n=1 (µ ∪ µ −1 n ) (x−1 ) = ea (x−1 ) ∨ ∨∞ ∪ µ−1 )n (x−1 ) = ea (x−1 ) ∨ ∪∞ n=1 (µ ∪ µ n=1 (µ −1 n −1 n (x) = ν(x). ) (x) = ea ∪ ∪∞ ) = ea (x) ∨ ∨∞ n=1 (µ ∪ µ n=1 (µ ∪ µ Let x, y ∈ G and let n ∈ N be such that n ≥ 2. Then (µ ∪ µ−1 )n (xy) = ∨{(µ ∪ µ−1 )(νn ) ∧ · · · ∧ (µ ∪ µ−1 )(ν1 )|xy = νn · · · ν1 , νi ∈ G, i = 1, . . . , n} ≥ ∨{(µ ∪ µ−1 )(x1 ) ∧ · · · ∧ (µ ∪ µ−1 )(xk ) ∧ (µ ∪ µ−1 )(yk+1 )∧· · ·∧(µ∪µ−1 )(yn )}|x = x1 . . . xk , y = yk+1 . . . yn , xi , yj ∈ G, i = 1, . . . , k, j = k + 1, . . . , n; k ∈ {1, 2, . . . , n − 1}} ≥ ∨{(µ ∪ µ−1 )(x1 ) ∧ · · · ∧ (µ ∪ µ)−1 (xk )|x = x1 . . . xk , xi ∈ G, i = 1, . . . , k} ∧ ∨{(µ ∪ µ−1 )(yk+1 ) ∧ · · · ∧ (µ ∪ µ−1 )(yn )|y = yk+1 . . . yn , yj ∈ G, j : k + 1, . . . , n} = (µ ∪ µ−1 )k (x) ∧ (µ ∪ µ−1 )n−k (y) for k ∈ {1, . . . , n − 1}. Hence (µ ∪ µ−1 )n (xy) ≥ (x) ∧ (µ ∪ µ−1 )n−k , n − 1. (µ ∪ µ−1 )k (y), k = 1, . . .−1 ∞ −1 n (xy) ≥ (µ ∪ µ (µ ∪ µ ) )k (x) ∧ (µ ∪ µ−1 )n−k (y), ∀k = Thus n=1 1, . .. , n − 1. Hence ∞ ∞ ∞ −1 n −1 n −1 n ) (xy) ≥ ) (x) ∧ ) (y). n=1 (µ ∪ µ n=1 (µ ∪ µ n=1 (µ ∪ µ Thus it follows that ν is a fuzzy subgroup of G. Clearly, µ ⊆ ν. Hence µ ⊆ ν. Let ξ be a fuzzy subgroup of G such that µ ⊆ ξ. Then ea ⊆ ξ and (µ ∪ µ−1 )n ⊆ ξ. Thus ν ⊆ ξ. Hence ν ⊆ µ.

1.3 Normal Fuzzy Subgroups The notion of a normal subgroup is one of the central concepts of classical group theory. It plays an important role in the study of the general structure of groups. Just as a normal subgroup plays an important role in the classical group theory, a normal fuzzy subgroup plays a similar role in the theory of fuzzy subgroups. Theorem 1.3.1. Let µ ∈ FP(G). Then the following assertions are equivalent: (1) µ(yx) = µ(xy) ∀ x, y ∈ G; in this case, µ is called an Abelian fuzzy subset of G. (2) µ(xyx−1 ) = µ(y) ∀ x, y ∈ G. (3) µ(xyx−1 ) ≥ µ(y) ∀ x, y ∈ G. (4) µ(xyx−1 ) ≤ µ(y) ∀ x, y ∈ G. (5) µ ◦ ν = ν ◦ µ ∀ ν ∈ FP(G). Proof. (1) ⇒ (2): Let x, y ∈ G. Then µ(xyx−1 ) = µ(x−1 · xy) = µ(y). (2) ⇒ (3): Immediate. (3) ⇒ (4): µ(xyx−1 ) ≤ µ(x−1 · xyx−1 · (x−1 )−1 ) = µ(y) ∀ x, y ∈ G.

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1 Fuzzy Subsets and Fuzzy Subgroups

(4) ⇒ (1): Let x, y ∈ G. Then µ(xy) = µ(x · yx · x−1 ) ≤ µ(yx) = µ(y · xy · y −1 ) ≤ µ(xy). Hence µ(xy) = µ(yx). (1) ⇒ (5): Let x ∈ G. Then (µ ◦ ν)(x) = ∨y∈G {µ(xy −1 ) ∧ ν(y)} = ∨y∈G {µ(y −1 x) ∧ ν(y)} = (ν ◦ µ)(x). Hence µ ◦ ν = ν ◦ µ. (5) ⇒ (1): Now 1{y−1 } ◦ µ = µ ◦ 1{y−1 } ∀y ∈ G. Thus (1{y−1 } ◦ µ)(x) = (µ ◦ 1{y−1 } )(x) ∀x, y ∈ G. Hence µ(yx) = µ(xy) ∀ x, y ∈ G. Deﬁnition 1.3.2. Let µ ∈ F(G). Then µ is called a normal fuzzy subgroup of G if it is an Abelian fuzzy subset of G. Let N F(G) denote the set of all normal fuzzy subgroups of G. If µ, ν ∈ F(G) and there exists u ∈ G such that µ(x) = ν(uxu−1 ) ∀x ∈ G, then µ and ν are called conjugate fuzzy subgroups (with respect to u) and we write, µ = ν u , where ν u (x) = ν(uxu−1 ) for all x ∈ G. Clearly, 1G and 1{e} are normal fuzzy subgroups of G. If G is a commutative group, every fuzzy subgroup of G is normal. A fuzzy subgroup µ of G is normal if and only if µ = µz ∀z ∈ G. Theorem 1.3.3. Let µ ∈ FP(G). Then µ ∈ N F(G) if and only if µa is a normal subgroup of G ∀a ∈ µ(G) ∪ {b ∈ [0, 1] | b ≤ µ(e)}. Proof. Suppose that µ ∈ N F(G). Let a ∈ µ(G) ∪ {b ∈ [0, 1] | b ≤ µ(e)}. Since µ ∈ F(G), µa is a subgroup of G. If x ∈ G and y ∈ µa , it follows from Theorem 1.3.1 that µ(xyx−1 ) = µ(y) ≥ a. Thus xyx−1 ∈ µa . Hence µa is a normal subgroup of G. Conversely, assume that µa is a normal subgroup of G ∀a ∈ µ(G) ∪ {b ∈ L | b ≤ µ(e)}. By Lemma 1.2.6, we have that µ ∈ F(G). Let x, y ∈ G and a = µ(y). Then y ∈ µa and so xyx−1 ∈ µa . Hence µ(xyx−1 ) ≥ a = µ(y). That is, µ satisﬁes condition (3) of Theorem 1.3.1. Consequently, it follows from Theorem 1.3.1 that µ ∈ N F(G). Theorem 1.3.4. Let µ ∈ N F(G). Then µ∗ and µ∗ are normal subgroups of G. Proof. Since µ ∈ F(G), it follows from Lemma 1.2.6 that µ∗ and µ∗ are subgroups of G. Let x ∈ G and y ∈ µ∗ . Since µ satisﬁes condition (2) of Theorem 1.3.1, we have µ(xyx−1 ) = µ(y) = µ(e) and thus xyx−1 ∈ µ∗ . Hence

1.3 Normal Fuzzy Subgroups

11

µ∗ is a normal subgroup of G. Let x ∈ G and y ∈ µ∗ . Since µ satisﬁes condition (2) of Theorem 1.3.1, it follows that µ(xyx−1 ) = µ(y) > 0 and so xyx−1 ∈ µ∗ . Therefore, µ∗ is a normal subgroup of G. Example 1.3.5. The converse of Theorem 1.3.4 is not true as can be seen by the following example: Let G be a group and H be a subgroup of G which is not normal. Deﬁne the fuzzy subset µ of G by µ(e) = 1, µ(x) = 12 if x ∈ H \ {e}, and µ(x) = 14 if x ∈ G \ H. Then µ is a fuzzy subgroup of G since its level sets are subgroups of G. Now µ 21 = H is not normal in G. Hence µ is not a normal fuzzy subgroup of G. However µ∗ = {e} and µ∗ = G are normal in G. The proof of the following result can be found in [[52], Theorem 5.2.6, p. 122] and Chapter 9. We state it here for information purposes. Theorem 1.3.6. Let a ∈ (0, 1] and let N F a (G) = {µ ∈ N F(G) | µ(e) = a}. The set N F a (G) together with ⊆ constitutes a complete lattice whose meet is fuzzy subset intersection ∩ and whose join is the product ◦. Moreover, N F a (G) is closed under intersection. Theorem 1.3.7. Suppose µ ∈ F(G). Let N (µ) = {x | x ∈ G, µ(xy) = µ(yx) ∀ y ∈ G}. Then N (µ) is a subgroup of G and the restriction of µ to N (µ), µ|N (µ) , is a normal fuzzy subgroup of N (µ). Proof. Clearly, e ∈ N (µ). Let x, y ∈ N (µ). For any z ∈ G, we see that µ(xy −1 · z) = µ(x · y −1 z) = µ(y −1 z · x) = µ(x−1 z −1 · y) = µ(y · x−1 z −1 ) = µ(z · xy −1 ). Thus xy −1 ∈ N (µ). Hence N (µ) is a subgroup of G. By Comment 1.2.4, it follows that µ|N (µ) ∈ F(N (µ)) and µ|N (µ) (xy) = µ|N (µ) (yx) ∀ x, y ∈ N (µ). Therefore, µ|N (µ) ∈ N F(N (µ)). The subgroup N (µ) of G deﬁned in Theorem 1.3.7 is called the normalizer of µ in G. Theorem 1.3.8. Let ν ∈ F(G). Then the cardinal number of the set {ν u | u ∈ G} is equal to the index [G : N (ν)] of the normalizer N (ν) in G. Proof. Let u, v ∈ G. Then ν u = ν v ⇔ ν(uxu−1 ) = ν(vxv −1 ) ∀ x ∈ G ⇔ ν(uv −1 · x) = ν(x · uv −1 ) ∀ x ∈ G ⇔ uv −1 ∈ N (ν) ⇔ u−1 N (ν) = v −1 N (ν). Thus ν u → u−1 N (ν) is a bijection from {ν u | u ∈ G} onto {uN (ν) | u ∈ G}. Theorem 1.3.9. Let ν ∈ F(G). Then ∩u∈G ν u ∈ N F(G) and is the largest normal fuzzy subgroup of G that is contained in ν.

12

1 Fuzzy Subsets and Fuzzy Subgroups

Proof. Since ν u ∈ F(G) ∀ u ∈ G, ∩u∈G ν u ∈ F(G). For all x ∈ G, observe that {ν u | u ∈ G} = {ν ux | u ∈ G}. Thus ∧u∈G ν u (xyx−1 ) = ∧u∈G ν(uxy(ux)−1 ) = ∧u∈G ν ux (y) = ∧u∈G ν u (y) ∀ x, y ∈ G. Hence by Theorem 1.3.1, ∩u∈G ν u ∈ N F(G). Now let µ ∈ N F(G) with µ ⊆ ν. Then µ = µu ⊆ ν u ∀ u ∈ G. Thus µ ⊆ ∩u∈G ν u . Therefore, ∩u∈G ν u is the largest normal fuzzy subgroup of G that is contained in ν. Let µ ∈ F(G) and x ∈ G. The fuzzy subsets µ(e){x} ◦ µ and µ ◦ µ(e){x} are referred to as the left coset and right coset of µ with respect to x, and written as xµ and µx, respectively. From Theorem 1.3.1, we know that if µ ∈ N F(G), then the left coset xµ is just the right coset µx. Thus in this case, we call xµ a coset for short. Theorem 1.3.10. Let µ ∈ F(G). Then for all x, y ∈ G, (1) xµ = yµ ⇔ xµ∗ = yµ∗ ; (2) µx = µy ⇔ µ∗ x = µ∗ y. Proof. Suppose that xµ = yµ. Then µ(e){x} ◦ µ = µ(e){y} ◦ µ which means that µ(x−1 z) = µ(y −1 z) ∀ z ∈ G. Choosing z = y yields µ(x−1 y) = µ(y −1 y) = µ(e) and thus x−1 y ∈ µ∗ . Hence xµ∗ = yµ∗ . Conversely, suppose that xµ∗ = yµ∗ . Then x−1 y ∈ µ∗ and y −1 x ∈ µ∗ . Hence µ(x−1 z) = µ(x−1 y · y −1 z) ≥ µ(x−1 y) ∧ µ(y −1 z) = µ(e) ∧ µ(y −1 z) = µ(y −1 z) ∀ z ∈ G. Similarly, µ(y −1 z) ≥ µ(x−1 z) ∀z ∈ G. Therefore, µ(x−1 z) = µ(y −1 z) ∀z ∈ G which shows that xµ = yµ. Similar reasoning shows that (2) holds. Theorem 1.3.11. Let µ ∈ N F(G) and x, y ∈ G. If xµ = yµ, then µ(x) = µ(y). Proof. Suppose that xµ = yµ. By Theorem 1.3.10, x−1 y ∈ µ∗ and y −1 x ∈ µ∗ . Since µ ∈ N F(G) it follows that µ(x) = µ(y −1 xy) ≥ µ(y −1 x) ∧ µ(y) = µ(e) ∧ µ(y) = µ(y). Similarly, µ(y) ≥ µ(x) and therefore µ(x) = µ(y). Theorem 1.3.12. Let µ ∈ N F(G). Set G/µ = {xµ | x ∈ G}. Then the following assertions hold: (1) (xµ) ◦ (yµ) = (xy)µ ∀ x, y ∈ G. (2) (G/µ, ◦) is a group. (3) G/µ = G/µ∗ . (4) Let µ(∗) ∈ FP(G/µ) be deﬁned by µ(∗) (xµ) = µ(x) ∀ x ∈ G. Then µ(∗) ∈ N F(G/µ).

1.3 Normal Fuzzy Subgroups

13

Proof. (1) For all x, y ∈ G, we have (xµ) ◦ (yµ) = (µ(e){x} ◦ µ) ◦ (µ(e){y} ◦ µ) = µ(e){x} ◦ (µ ◦ µ(e){y} ) ◦ µ = µ(e){x} ◦ (µ ◦ µ) ◦ µ(e){y} = µ(e){x} ◦ µ ◦ µ(e){y} = µ(e){x} ◦ (µ ◦ µ(e){y} ) = µ(e){x} ◦ (µ(e){y} ◦ µ) = (µ(e){x} ◦ µ(e){y} ) ◦ µ = (xy)µ by Theorems 1.2.2 and 1.3.1. (2) By (1), G/µ is closed under the operation ◦. Also, ◦ satisﬁes the associative law. Now µ ◦ (xµ) = (eµ) ◦ (xµ) = (ex)µ = xµ ∀x ∈ G and

(x−1 µ) ◦ (xµ) = (x−1 x)µ = eµ = µ ∀x ∈ G.

Hence (G/µ, ◦) is a group. (3) Since µ ∈ N F(G), it follows by Theorem 1.3.4 that µ∗ is a normal subgroup of G. Hence (G/µ∗ ) is a group and the function f : (G/µ) → (G/µ∗ ) given by xµ → xµ∗ is an isomorphism by Theorem 1.3.10 and the fact that xµ ◦ yµ = (xy)µ and xµ∗ yµ∗ = (xy)µ∗ . (4) By Theorem 1.3.11, xµ = yµ implies that µ(x) = µ(y). Thus µ(∗) is single-valued. Since µ(∗) ((xµ)−1 ) = µ(∗) (x−1 µ) = µ(x−1 ) = µ(x) = µ(∗) (xµ) ∀ x ∈ G and

µ(∗) ((xµ) ◦ (yµ)) = µ(∗) (xyµ) = µ(xy) ≥ µ(x) ∧ µ(y) = µ(∗) (xµ) ∧ µ(∗) (yµ)

∀ x, y ∈ G, it follows that µ(∗) ∈ F(G/µ). Moreover, since µ(∗) ((xµ) ◦ (yµ)) = µ(∗) (xyµ) = µ(xy) = µ(yx) = µ(∗) (yxµ) = µ(∗) ((yµ) ◦ (xµ)) ∀ x, y ∈ G, we have that µ(∗) ∈ N F(G/µ).

The group G/µ deﬁned in Theorem 1.3.12 is called the quotient group (or factor group) of G relative to the normal fuzzy subgroup µ.

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1 Fuzzy Subsets and Fuzzy Subgroups

Theorem 1.3.13. Let ν ∈ F(G) and let N be a normal subgroup of G. Deﬁne ξ ∈ FP(G/N ) as follows: ξ(xN ) = ∨{ν(z) | z ∈ xN } ∀x ∈ G. Then ξ ∈ F(G/N ). Proof. Now

ξ((xN )−1 ) = ξ(x−1 N ) = ∨{ν(z) | z ∈ x−1 N } = ∨{ν(w−1 ) | w−1 ∈ x−1 N } = ∨{ν(w) | w ∈ xN } = ξ(xN )

∀ x ∈ G; ξ(xN yN ) = ∨{ν(z) | z ∈ xyN } = ∨{ν(uv) | u ∈ xN, v ∈ yN } ≥ ∨{ν(u) ∧ ν(v) | u ∈ xN, v ∈ yN } = (∨{ν(u) | u ∈ xN }) ∧ (∨{ν(v) | v ∈ yN }) = ξ(xN ) ∧ ξ(yN ) ∀ x, y ∈ G. Hence ξ ∈ F(G/N ).

The fuzzy subgroup ξ deﬁned in Theorem 1.3.13 is called the quotient fuzzy subgroup (or factor fuzzy subgroup) of the fuzzy subgroup ν of G relative to the normal subgroup N of G and is denoted by ν/N. Theorem 1.3.14. Let µ ∈ N F(G) and H be a group. Suppose that f is an epimorphism of G onto H. Then f (µ) ∈ N F(H). Proof. By Theorem 1.2.11, f (µ) ∈ F(H). Now let x, y ∈ H. Since f is a surjection, f (u) = x for some u ∈ G. Thus f (µ)(xyx−1 ) = ∨{µ(w) | w ∈ G, f (w) = xyx−1 } = ∨{µ(u−1 wu) | w ∈ G, f (u−1 wu) = y} = ∨{µ(w) | uwu−1 ∈ G, f (w) = y} = ∨{µ(w) | w ∈ G, f (w) = y} = f (µ)(y). Therefore, it follows from Theorem 1.3.1 that f (µ) ∈ N F(H).

Theorem 1.3.15. Let H be a group and ν ∈ N F(H). If f is a homomorphism from G into H, then f −1 (ν) ∈ N F(G). Proof. By Theorem 1.2.12, f −1 (ν) ∈ F(G). Now for any x, y ∈ G, we have f −1 (ν)(xy) = ν(f (xy)) = ν(f (x)f (y)) = ν(f (y)f (x)) = ν(f (yx)) = f −1 (ν)(yx). Hence f −1 (ν) ∈ N F(G).

1.4 Homomorphisms and Isomorphisms

15

1.4 Homomorphisms and Isomorphisms This section consists of two parts. The ﬁrst part is mainly concerned with a generalization of the concept of a normal fuzzy subgroup. We introduce the concept of a normal fuzzy subgroup of a fuzzy subgroup and examine its basic properties. In the second part, we introduce the concepts of homomorphisms and isomorphisms of fuzzy subgroups and use them to develop results concerning fuzzy subgroups of a fuzzy subgroup. Deﬁnition 1.4.1. Let µ, ν ∈ F(G) and µ ⊆ ν. Then µ is called a normal fuzzy subgroup of the fuzzy subgroup ν, written µ ν, if µ(xyx−1 ) ≥ µ(y) ∧ ν(x) ∀x, y ∈ G. The following statements are immediate from Deﬁnition 1.4.1. (1) If G1 and G2 are subgroups of G, then G1 is a normal subgroup of G2 if and only if 1G1 is a normal fuzzy subgroup of 1G2 . (2) If µ ∈ N F(G), ν ∈ F(G), and µ ⊆ ν, then µ is a normal fuzzy subgroup of ν. (3) Every fuzzy subgroup is a normal fuzzy subgroup of itself. (4) µ ∈ FP(G) is a normal fuzzy subgroup of G if and only if µ is a normal fuzzy subgroup of the fuzzy subgroup 1G . Theorem 1.4.2. Let µ, ν ∈ F(G) and µ ⊆ ν. Then the following assertions are equivalent: (1) µ is a normal fuzzy subgroup of ν. (2) µ(yx) ≥ µ(xy) ∧ ν(y) ∀ x, y ∈ G. (3) µ(e){x} ◦ µ ⊇ (µ ◦ µ(e){x} ) ∩ ν ∀ x ∈ G. Proof. (1) ⇒ (2): Since µ is a normal fuzzy subgroup of ν, it follows that µ(yx) = µ(yxyy −1 ) ≥ µ(xy) ∧ ν(y) ∀x, y ∈ G. (2) ⇒ (3): For any z ∈ G, (µ(e){x} ◦ µ)(z) = ∨{[µ(e){x} (p) ∧ µ(q)]|pq = z} = ∨{[µ(e){x} (x) ∧ µ(q)]|xq = z} = ∨[µ(e) ∧ µ(x−1 z)] = ∨[µ(x−1 z)] = ∨µ(z −1 x) ≥ ∨µ(xz −1 ) ∧ ν(z −1 ) = ∨µ(zx−1 ) ∧ ν(z −1 ) = (µ ◦ µ(e){x} )(z) ∧ ν(z) = ((µ ◦ µ(e){x} ) ∩ ν)(z). (2) ⇒ (1):

µ(xyx−1 ) ≥ µ(yx−1 x) ∧ ν(x) = µ(y) ∧ ν(x).

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1 Fuzzy Subsets and Fuzzy Subgroups

(3) ⇒ (2): ∀ x, y ∈ G, µ(yx) = µ(x−1 y −1 ) = (µ(e){x} ◦ µ)(y −1 ) ≥ ((µ ◦ µ(e){x} ) ∩ ν)(y −1 ) = µ(y −1 x−1 ) ∧ ν(y −1 ) = µ(xy) ∧ ν(y). Theorem 1.4.3. Let µ, ν ∈ F(G). Then µ is a normal fuzzy subgroup of ν if and only if µa is a normal subgroup of νa ∀a ∈ {b ∈ [0, 1] | b ≤ µ(e)}. Proof. Suppose µ is a normal fuzzy subgroup of ν. Let a ∈ {b ∈ [0, 1]|b ≤ µ(e)}. Then µa is a subgroup of νa . Let y ∈ µa and x ∈ νa . Then µ(y) ≥ a and ν(x) ≥ a. By hypothesis, µ(xyx−1 ) ≥ µ(y) ∧ ν(x) ≥ a ∧ a = a. Hence xyx−1 ∈ µa . That is, µa is a normal subgroup of νa . Conversely suppose µa is a normal subgroup of νa for all a ∈ {b ∈ [0, 1]|b ≤ µ(e)}. Let µ(y) = a, ν(x) = b and suppose that b ≥ a. Then x ∈ νa and so xyx−1 ∈ µa by hypothesis. Thus µ(xyx−1 ) ≥ a = a ∧ b = µ(y) ∧ ν(x). Suppose b < a. Then y ∈ µb . Since µb is a normal subgroup of νb , this implies xyx−1 ∈ µb . Thus, µ(xyx−1 ) ≥ b = b ∧ a = ν(x) ∧ µ(y). Theorem 1.4.4. Let µ, ν ∈ F(G) and µ be a normal fuzzy subgroup of ν. Then µ∗ is a normal subgroup of ν∗ and µ∗ is a normal subgroup of ν ∗ . Proof. If x ∈ µ∗ (respectively x ∈ µ∗ ) and y ∈ ν∗ (respectively y ∈ ν ∗ ), then µ a normal fuzzy subgroup of ν implies that µ(y −1 xy) ≥ (µ(x) ∧ ν(y)) = µ(e) ∧ ν(e) = µ(e) (respectively µ(y −1 xy) > 0). This shows µ∗ is a normal subgroup of ν∗ (respectively µ∗ is a normal subgroup of ν ∗ ). Theorem 1.4.5. If µ ∈ N F(G) and ν ∈ F(G), then µ ∩ ν is a normal fuzzy subgroup of ν. Proof. Clearly, µ ∩ ν ∈ F(G) and µ ∩ ν ⊆ ν. Now (µ ∩ ν)(xyx−1 ) = µ(xyx−1 ) ∧ ν(xyx−1 ) = µ(y) ∧ ν(xyx−1 ) ≥ µ(y) ∧ ν(x) ∧ ν(y) ∧ ν(x−1 ) = (µ ∩ ν)(y) ∧ ν(x) ∀ x, y ∈ G. Hence µ ∩ ν is a normal fuzzy subgroup of ν.

Theorem 1.4.6. Let µ, v, ξ ∈ F(G) be such that µ and ν are normal fuzzy subgroups of ξ. Then µ ∩ ν is a normal fuzzy subgroup of ξ. Proof. Observe that µ ∩ ν ∈ F(G) and µ ∩ ν ⊆ ξ. Now (µ ∩ ν)(xyx−1 ) = µ(xyx−1 ) ∧ ν(xyx−1 ) ≥ (µ(y) ∧ ξ(x)) ∧ (ν(y) ∧ ξ(x)) ≥ (µ ∩ ν)(y) ∧ ξ(x). Therefore, µ ∩ ν is a normal fuzzy subgroup of ξ.

1.4 Homomorphisms and Isomorphisms

17

Theorem 1.4.7. Let µ, ν ∈ F(G) and µ be a normal fuzzy subgroup of ν. Let H be a group and f a homomorphism from G into H. Then f (µ) is a normal fuzzy subgroup of f (ν). Proof. Note, f (µ), f (ν) ∈ F(H) and f (µ) ⊆ f (ν). Now (f (µ))(xyx−1 ) = ∨{µ(z) | z ∈ G, f (z) = xyx−1 } ≥ ∨{µ(uvu−1 ) | u, v ∈ G, f (u) = x, f (v) = y} ≥ ∨{µ(v) ∧ ν(u) | u, v ∈ G, f (u) = x, f (v) = y} = (∨{µ(v) | v ∈ G, f (v) = y}) ∧ (∨{ν(u) | u ∈ G, f (u) = x}) = (f (µ))(y) ∧ (f (ν))(x) ∀ x, y ∈ H. Hence f (µ) is a normal fuzzy subgroup of f (ν).

Theorem 1.4.8. Let H be a group. Let µ, ν ∈ F(H) and µ be a normal fuzzy subgroup of ν. Let f be a homomorphism from G into H. Then f −1 (µ) is a normal fuzzy subgroup of f −1 (ν). Proof. Clearly, f −1 (µ), f −1 (ν) ∈ F(G). It follows easily that f −1 (µ) ⊆ f −1 (ν). Now f −1 (µ)(xyx−1 ) = µ(f (xyx−1 )) = µ(f (x)f (y)(f (x))−1 ) ≥ µ(f (y)) ∧ ν(f (x)) = f −1 (µ)(y) ∧ (f −1 (ν))(x) ∀ x, y ∈ G. Hence f −1 (µ) is a normal fuzzy subgroup of f −1 (ν).

Deﬁnition 1.4.9. Let G and H be groups and let µ ∈ F(G) and ν ∈ F(H). (1) A homomorphism f of G onto H is called a weak homomorphism of µ into ν if f (µ) ⊆ ν. If f is a weak homomorphism of µ into ν, then we say f

that µ is weakly homomorphic to ν and we write µ ∼ ν, or simply µ ∼ ν. (2) An isomorphism f of G onto H is called a weak isomorphism of µ into ν if f (µ) ⊆ ν. If f is a weak isomorphism of µ into ν, then we say that f

µ is weakly isomorphic to ν and we write µ ν, or simply µ ν. (3) A homomorphism f of G onto H is called a homomorphism of µ onto ν if f (µ) = ν. If f is a homomorphism of µ onto ν, then we say that µ f

is homomorphic to ν and we write µ ≈ ν, or simply µ ≈ ν. (4) An isomorphism f of G onto H is called an isomorphism of µ onto ν if f (µ) = ν. If f is an isomorphism of µ onto ν, then we say that µ is f

isomorphic to ν and we write µ = ν, or simply µ = ν. Let µ, ν ∈ F(G). Suppose that µ is a normal fuzzy subgroup of ν. Then µ∗ is a normal subgroup of ν ∗ by Theorem 1.4.4. Clearly, ν|ν ∗ is a fuzzy subgroup of ν ∗ . Thus by Theorem 1.3.13, the factor fuzzy subgroup of ν|ν ∗ relative to µ∗ exists. For convenience sake, we denote this factor fuzzy subgroup by ν/µ and call it the quotient subgroup (or factor subgroup of ν relative to µ.

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1 Fuzzy Subsets and Fuzzy Subgroups

Theorem 1.4.10. Let µ, ν ∈ F(G) and µ be a normal fuzzy subgroup of ν. Then ν|ν ∗ ≈ ν/µ. Proof. Let f be the natural homomorphism from ν ∗ onto ν ∗ /µ∗ . Then f (ν|ν ∗ )(xµ∗ ) = ∨{ν|ν ∗ (z) | z ∈ ν ∗ , f (z) = xµ∗ } = ∨{ν(y) | y ∈ xµ∗ } = (ν/µ)(xµ∗ ) f

∀ x ∈ ν ∗ . Therefore, ν|ν ∗ ≈ ν/µ.

Lemma 1.4.11. If f : X → Y and µ ∈ FP(G), then (f (µ))∗ = f ((µ)∗ ) Theorem 1.4.12. Let ν ∈ F(G). Suppose that H is a group and ξ ∈ F(H) is such that ν ≈ ξ. Then there exists a normal fuzzy subgroup µ of ν such that ν/µ = ξ|ξ∗ . Proof. Since ν ≈ ξ, there exists an epimorphism f of G onto H such that f (ν) = ξ. Deﬁne µ ∈ FP(G) as follows: ∀ x ∈ G, ν(x) if x ∈ Ker f µ(x) = 0 otherwise. Clearly, µ ∈ F(G) and µ ⊆ ν. If x ∈ Ker f, then yxy −1 ∈ Ker f ∀ y ∈ G, and so µ(yxy −1 ) = ν(yxy −1 ) ≥ ν(x) ∧ ν(y) = µ(x) ∧ ν(y) ∀y ∈ G. If x ∈ G\Ker f, then µ(x) = 0 and so µ(yxy −1 ) ≥ µ(x) ∧ ν(y) ∀y ∈ G. f

Hence µ is a normal fuzzy subgroup of ν. Also, ν ≈ ξ implies f (ν) = ξ which further implies (f (ν))∗ = ξ ∗ . By Lemma 1.4.11 it follows that f (ν ∗ ) = ξ ∗ . Let g = f |ν ∗ . Then g is a homomorphism of ν ∗ onto ξ ∗ and Ker g = µ∗ by the deﬁnition of µ. Thus there exists an isomorphism h of ν ∗ /µ∗ onto ξ ∗ such that h(xµ∗ ) = g(x) = f (x) ∀ x ∈ ν ∗ . For such an h, we have h(ν/µ)(z) = ∨{(ν/µ)(xµ∗ ) | x ∈ ν ∗ , h(xµ∗ ) = z} = ∨{∨{ν(y) | y ∈ xµ∗ } | x ∈ ν ∗ , g(x) = z} = ∨{ν(y) | y ∈ ν ∗ , g(y) = z} = ∨{ν(y) | y ∈ G, f (y) = z} = ξ(z) h

∀ z ∈ ξ ∗ . Therefore, ν/µ = ξ|ξ∗ .

Theorem 1.4.13. Let µ ∈ N F(G) and ν ∈ F(G) be such that µ(e) = ν(e). Then ν/(µ ∩ ν) (µ ◦ ν)/µ.

1.4 Homomorphisms and Isomorphisms

19

Proof. From Theorem 1.3.4, we have that µ∗ is a normal subgroup of G. By the Second Isomorphism Theorem for Groups, ν ∗ /(µ∗ ∩ ν ∗ )=(µ ∗ ν ∗ )/µ∗ . One can verify that

(µ ∩ ν)∗ = µ∗ ∩ ν ∗ , (µ ◦ v)∗ = µ∗ ν ∗ .

Consequently, we have f

ν ∗ /(µ ∩ ν)∗ =(µ ◦ ν)∗ /µ∗ , where f is given by

f (x(µ ∩ ν)∗ ) = xµ∗

∀ x ∈ ν ∗ . Thus (since f is one-to-one) f (ν/(µ ∩ ν))(yµ∗ ) = (ν/(µ ∩ ν))(y(µ ∩ ν)∗ ) = ∨{ν(z) | z ∈ y(µ ∩ ν)∗ } ≤ ∨{(µ ◦ ν)(z) | z ∈ y(µ∗ ∩ ν ∗ )} (since µ(e) = ν(e)) ≤ ∨{(µ ◦ ν)(z) | z ∈ yµ∗ } = ((µ ◦ ν)/µ)(yµ∗ ) f

∀ y ∈ ν ∗ . Hence f (ν/(µ ∩ ν)) ⊆ (µ ◦ ν)/µ. Therefore, ν/(µ ∩ ν) (µ ◦ ν)/µ.

The next lemma can be proved using Theorem 1.4.10. Lemma 1.4.14. Let µ, ν, ξ ∈ F(G) be such that µ and ν are normal fuzzy subgroups of ξ and µ ⊆ ν, then (ν/µ) is a normal fuzzy subgroup of (ξ/ν). Theorem 1.4.15. Let µ, ν, ξ ∈ F(G) be such that µ ⊆ ν and µ and ν be normal fuzzy subgroups of ξ. Then (ξ/µ)/(ν/µ)=ξ/ν. Proof. By Theorem 1.4.4, µ∗ is a normal subgroup of ν ∗ , and both µ∗ and ν ∗ are normal subgroups of ξ ∗ . By the Third Isomorphism Theorem for Groups, f

(ξ ∗ /µ∗ )/(ν ∗ /µ∗ )=ξ ∗ /ν ∗ , where f is given by f (xµ∗ · (ν ∗ /µ∗ )) = xν ∗ ∀x ∈ ξ ∗ .

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1 Fuzzy Subsets and Fuzzy Subgroups

f ((ξ/µ)/(ν/µ))(xν ∗ ) = ((ξ/µ)/(ν/µ))(xµ∗ · (ν ∗ /µ∗ )) = ∨{(ξ/µ)(yµ∗ ) | y ∈ ξ ∗ , yµ∗ ∈ xµ∗ · (ν ∗ /µ∗ )} = ∨{∨{ξ(z) | z ∈ yµ∗ } | y ∈ ξ ∗ , yµ∗ ∈ xµ∗ · (ν ∗ /µ∗ )} = ∨{ξ(z) | z ∈ ξ ∗ , zµ∗ ∈ xµ∗ · (ν ∗ /µ∗ )} = ∨{ξ(z) | z ∈ xµ∗ · (ν ∗ /µ∗ )} = ∨{ξ(z)|z ∈ ξ ∗ , f (z) ∈ xν ∗ } = (ξ/ν)(xν ∗ ) ∀ x ∈ ξ ∗ , where the ﬁrst equality holds since f is one-to-one. Hence f

(ξ/µ)/(ν/µ)=ξ/ν. We now deﬁne a concept that is used later. Deﬁnition 1.4.16. Let f be a homomorphism of a group G into a group H. Then a fuzzy subgroup µ of G is called f -invariant if for all x, y ∈ G, f (x) = f (y) implies µ(x) = µ(y). Remark 1.4.17. We note that µ being f -invariant is equivalent to the statement that ∀x ∈ G, f (µ)(f (x)) = µ(x): Suppose this latter condition holds. Suppose f (x) = f (y) ∀x, y ∈ G. Then f (µ)(f (x)) = f (µ)(f (y)) and so µ(x) = µ(y). Conversely suppose µ is f -invariant. Let x ∈ G. Then f (µ)(f (x)) ≥ µ(x). Suppose f (µ)(f (x)) > µ(x). Then there exists a y ∈ G such that µ(y) > µ(x) and f (x) = f (y), a contradiction. Thus f (µ)(f (x)) = µ(x).

1.5 Complete and Weak Direct Products In order to deﬁne the complete direct product of fuzzy subgroups, we ﬁrst review the concept of the complete direct product of groups. Let {Gi | i ∈ I} be a collection of groups and let ei denote the identity of Gi for all i ∈ I. Multiplication on the Cartesian product i∈I Gi is deﬁned by (xi )i∈I (yi )i∈I = (xi yi )i∈I , ∀(xi )i∈I , (yi )i∈I ∈ i∈I Gi . Then the Cartesian product i∈I Gi together with the multiplication just deﬁned form a group with identity (ei )i∈I . This group is called the complete direct product of the Gi ’s and is denoted by ∼ Gi . In particular, when I = {1, 2, . . . , n} (where n ∈ N \{1}), we regard i∈I the Cartesian product i∈I Gi as the set G1 × G2 × . . . × Gn = {(x1 , x2 , . . . , xn ) | xi ∈ Gi , i ∈ I}

1.5 Complete and Weak Direct Products

and denote the complete direct product It follows that if G =

∼

∼

∼

∼

∼

Gi by G1 ⊗ G2 ⊗ . . . ⊗ Gn .

i∈I

Gi and µi ∈ F(Gi ) for all i ∈ I, then

i∈I

F(G), where ∀(xi )i∈I

21

∼

µi ∈

i∈I

∼ ( µi )((xi )i∈I ) = ∧i∈I µi (xi ). i∈I

In this case, we call the fuzzy subgroup

∼

µi of G the complete direct

i∈I

product of the µi ’s. Unless otherwise speciﬁed, it is assumed that Gi is a group with identity ∼ ei for all i ∈ I, G = Gi , and so e = (ei )i∈I . i∈I

Theorem 1.5.1. Let µi ∈ N F(Gi ) ∀ i ∈ I. Then µ =

∼

µi ∈ N F(G).

i∈I

Proof. Now µ ∈ F(G). For all x = (xi )i∈I and y = (yi )i∈I in G, we have µ(xy) = ∧i∈I µi (xi yi ) = ∧i∈I µi (yi xi ) = µ(yx). Hence µ ∈ N F(G). We introduce the notion of a weak direct product of fuzzy subgroups. Let xi ∈ G for i ∈ I, where |I| > 1. Suppose that there are at most ﬁnitely many xi not equal to e. We deﬁne i∈I xi as follows: if xi is equal to e for x as e; and if xi = e for some i ∈ I, then every i ∈ I, then we interpret i i∈I x is merely the product of those xi that are not equal to e. i i∈I Note that if there are inﬁnitely many xi ’s not equal to e, then i∈I xi is undeﬁned. For the sake of convenience, we assume that i∈I xi is meaningful whenever it appears in the ensuing discussion. In the remainder of this section |I| > 1 and we write xi for i∈I xi . Clearly, every element x of G can be expressed in the form x = xi . Now let Gi be a submonoid of G for all i ∈ I and assume that xi ∈ Gi and xj ∈ Gj for i, j ∈ I with i = j ⇒ xi xj = xj xi . Then x=

xi and y =

yi , where xi , yi ∈ Gi ∀ i ∈ I ⇒ xy =

(xi yi ).

For subsets Ai (i ∈ I) of G, the set { xi | xi ∈ Ai , i ∈ I} is called the ∗ weak product of the Ai ’s and written as Ai . If e ∈ Ai ∀ i ∈ I, then Aj ⊆

∗ i∈I

i∈I

Ai ∀ j ∈ I.

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1 Fuzzy Subsets and Fuzzy Subgroups

For subgroups Gi of G (i ∈ I), the group G is called the weak direct · Gi , if the following conditions product of the Gi ’s, and written as G = i∈I

are satisﬁed: ∗ (1) G = Gi , i∈I

and xj ∈ Gj for i, j ∈ I with i = j ⇒ xi xj = xj xi , (2) x i ∈ Gi (3) xi = yi , where xi , yi ∈ Gi ⇒ xi = yi ∀ i ∈ I. It can be shown that condition (2) is equivalent to the following condition: (2) every Gi is a normal subgroup of G; and condition (3) is equivalent to either of the following two conditions: (3) e = xi , where xi ∈ Gi ⇒ xi = e ∀ i ∈ I; ∗ Gi ) = {e} ∀ j ∈ I. (3) Gj ∩ ( i∈I\{j}

∗

(If I \ {j} is a singleton {i0 }, we write

Gi = Gi0 .)

i∈I\{j}

Recall from group theory that if I is ﬁnite, then the complete direct product · and the weak direct product coincide. If G is the weak direct product Gi i∈I

and I is ﬁnite, say I = {1, 2, . . . , n}, then we sometimes use the notation ·

·

·

G1 ⊗ G2 ⊗ ... ⊗ Gn or the more standard notation G1 ⊗G2 ⊗ . . . ⊗Gn to denote the weak direct product of the Gi ’s, and call G the direct product of the Gi ’s. We now deﬁne the weak product of fuzzy subsets as follows. Deﬁnition 1.5.2. For all i ∈ I, let µi ∈ FP(G). Deﬁne µ ∈ FP(G) as follows: µ(x) = ∨{(∧i∈I µi (xi )) | xi ∈ G, i ∈ I, xi = x} ∀ x ∈ G. Then µ is called the weak product of the µi ’s and is denoted by µ=

∗

µi .

i∈I

Let I = {1, 2, . . . , n}, where n ∈ N \{1}. If µi (e) ≥ µj (x) and µi ⊆ µ ∀ i, j ∈ I and ∀x ∈ G, then ∗

µi ⊆ µ1 ◦ µ2 ◦ . . . ◦ µn .

i∈I

However, if the µi ’s also satisfy the following condition: xi ∈ µ∗i and xj ∈ µ∗j for i, j ∈ I with i = j ⇒ xi xj = xj xi , then

1.5 Complete and Weak Direct Products ∗

23

µi = µ1 ◦ µ2 ◦ . . . ◦ µn .

i∈I

Therefore, the product of ﬁnitely many fuzzy subsets of a commutative group can be treated as a special case of the weak product. ∗

Theorem 1.5.3. For all i ∈ I, let µi ∈ F(G) and let µ =

µi . Then the

i∈I

following assertions hold: ∗ (1) µ∗ ⊇ (µi )∗ . i∈I

(2) If ∨(( ∪i∈I µi (G)) \{µ(e)}) < µ(e), then µ∗ =

∗

(µi )∗ .

i∈I ∗

Theorem 1.5.4. For all i ∈ I, let µi ∈ FP(G) and let µ = µ∗ =

∗ i∈I

µi . Then

i∈I

µ∗i .

Let µ ∈ FP(G). Then µ is called a fuzzy subsemigroup of G if µ(xy) ≥ µ(x) ∧ µ(y) ∀x, y ∈ G and a fuzzy submonoid of G if in addition to being a fuzzy subsemigroup of G, µ(e) ≥ µ(x) ∀x ∈ G. Theorem 1.5.5. For all i ∈ I, let µi ∈ FP(G) and let µ =

∗

µi . Suppose

i∈I

that G is a commutative group. Then the following assertions hold: (1) If every µi is a fuzzy subsemigroup of G, then so is µ. (2) If every µi is an fuzzy submonoid of G and µi (e) = µj (e) ∀i, j ∈ I, then µ is the smallest fuzzy submonoid of G that contains all µi ’s. (3) If every µi is an fuzzy subgroup of G and µi (e) = µj (e) ∀i, j ∈ I, then µ is the smallest fuzzy subgroup of G that contains all µi ’s, that is, µ = ∪i∈I µi . Proof. (1) If every µi is a fuzzy subsemigroup of G, then µ(xy) = ∨{∧i∈I µi (zi ) | zi ∈ G, i ∈ I, zi = xy} ≥ ∨{∧i∈I µi (xi yi ) | xi , yi ∈ G, i ∈ I, xi = x, yi = y} ≥ ∨{∧i∈I (µi (xi ) ∧ µi (yi )) | xi , yi ∈ G, i ∈ I, xi = x, yi = y} (xi )) ∧ (∧i∈I µi (yi )) | xi , yi ∈ G, i ∈ I, ≥ ∨{(∧i∈I µi xi = x, yi = y} = (∨{∧i∈I µi (xi ) | xi ∈ G, i ∈ I, xi = x}) ∧(∨{∧i∈I µi (yi ) | yi ∈ G, i ∈ I, yi = y}) = µ(x) ∧ µ(y) ∀ x, y ∈ G. Hence µ is a fuzzy subsemigroup of G. (2) If every µi is a fuzzy submonoid of G, then µ is also a fuzzy subsemigroup G, and in addition we have

24

1 Fuzzy Subsets and Fuzzy Subgroups

µ(e) = ∧i∈I µi (e). Thus µ is a fuzzy submonoid of G. Now since µi ⊆ µ ∀i ∈ I and for any fuzzy submonoid ν of G that contains all µi ’s, x= xi , where x, xi ∈ G for i ∈ I ⇒ ∧i∈I µi (xi ) ≤ ∧i∈I ν(xi ) ≤ ν(x), it follows that µ ⊆ ν. Therefore, µ is the smallest fuzzy submonoid of G that contains all µi ’s. (3) If every µi is a fuzzy subgroup of G, then µ is a fuzzy submonoid of G. Also, yi = x−1 } µ(x−1 ) = ∨{∧i∈I µi (yi ) | yi ∈ G, i ∈ I, −1 = ∨{∧i∈I µi (yi ) | yi ∈ G, i ∈ I, yi−1 = x} = µ(x) ∀ x ∈ G. Thus µ is also a fuzzy subgroup of G. It has been shown that µ, being a fuzzy submonoid of G, is the smallest fuzzy submonoid of G that contains all µi ’s. Naturally, µ is also the smallest fuzzy subgroup of G that contains all µi ’s, that is, µ = ∪i∈I µi . In the preceding theorem, G is assumed to be a commutative group. For an arbitrary group, we have the following theorem. Theorem 1.5.6. For all i ∈ I, let µi ∈ FP(G) and µ =

∗

µi . Suppose that

i∈I

the µi ’s satisfy the following condition: xi ∈ µ∗i and xj ∈ µ∗j for i = j ⇒ xi xj = xj xi . If every µi is a fuzzy subsemigroup (fuzzy submonoid, fuzzy subgroup, normal fuzzy subgroup) of G with µi (e) = µj (e) ∀i, j ∈ I, then so is µ. Proof. The proofs of the ﬁrst three parts of the theorem are similar to those of Theorem 1.5.5 and thus are omitted. We now prove the last part. Assume that every µi is a normal fuzzy subgroup of G. By the third part, we conclude that µ is a fuzzy subgroup of G. Hence −1 µ(xyx−1 ) = ∨{∧i∈I µi (zi ) | zi ∈ G, i ∈ I, zi = xyx −1} −1 ∈ I, xyi x = xyx−1 } = ∨{∧i∈I µi (xyi x ) | yi ∈ G, i = ∨{∧i∈I µi (yi ) | yi ∈ G, i ∈ I, yi = y} = µ(y) ∀ x, y ∈ G. Thus µ is a normal fuzzy subgroup of G.

1.5 Complete and Weak Direct Products

25

Theorem 1.5.7. For all i ∈ I, let µi ∈ F(G) be such that µi (e) = µj (e) ∗ ∀i, j ∈ I. Let µ = µi . Suppose that H and Hi (i ∈ I) are subgroups of G i∈I

such that

µ∗i

⊆ Hi ∀ i ∈ I and H =

·

Hi , i.e., H is the weak direct product

i∈I

of the Hi ’s. Then the following assertions hold. (1) µ ∈ F(G). (2) Every µi is a normal fuzzy subgroup of the fuzzy subgroup µ. ∗ µi ) = µ(e){e} ∀ j ∈ I. (3) µj ∩ ( i∈I\{j}

(When I\{j} is a singleton {i0 }, we write

∗

µi = µi0 ).

i∈I\{j}

(4) If µi |H ∈ N F(H) ∀ i ∈ I, then µ|H ∈ N F(H). Proof. Let x, y ∈ G and consider µ(xy). For either x ∈ / H or y ∈ / H, clearly µ(xy) ≥ 0 = µ(x) ∧ µ(y). If x, y ∈ H, we have xy ∈ H and the proof that (1) holds follows from Theorem 1.5.5. Now we show assertion (2) holds. Let j ∈ I. Since µj ∈ F(G) and µj ⊆ µ, in view of part (2) of Theorem 1.4.2, it needs only to be shown that µj (yx) ≥ µj (xy) ∧ µ(y) ∀x, y ∈ G. If either xy ∈ / Hj or y ∈ / H, it is clear that µj (yx) ≥ 0 = µj (xy) ∧ µ(y). Now assume that xy ∈ Hj and y ∈ H. Since Hj ⊆ H, we have x ∈ H. Also, from the fact that Hj is a normal subgroup of H, it follows that yx = x−1 ·xy·x ∈ Hj . Thus using the fact that xy, yx ∈ Hj , we have that µ(xy) = µj (xy) and µ(yx) = µj (yx). Hence µj (yx) = µ(yx) = µ(y · xy · y −1 ) ≥ µ(xy) ∧ µ(y) = µj (xy) ∧ µ(y). Therefore, µj is a normal fuzzy subgroup of µ, that is, assertion (2) holds. It now follows that (3) holds. To show (4), we suppose that µi |H ∈ N F(H) ∀ i ∈ I. Clearly, µ| H ∈ F(H). Next, for any x, y ∈ H, x and y have a unique expression as x = xi and y = y , where x , y ∈ H ∀ i ∈ I. Hence xy = (x y ) and yx = i i i i i i (yi xi ). Thus µ(xy) = ∧i∈I µi (xi yi ) = ∧i∈I µi (yi xi ) = µ(yx). Therefore, µ|H ∈ N F(H). That is, assertion (4) holds.

26

1 Fuzzy Subsets and Fuzzy Subgroups

Further, it is easy to show that µ|H in Theorem 1.5.7 is isomorphic to ∼ some fuzzy subgroup of the group Hi . i∈I

Deﬁnition 1.5.8. Let µ ∈ F(G) and µi ∈ F(G) for all i ∈ I. Suppose that µi (e) = µj (e) ∀i, j ∈ I. Then µ is called the weak direct product of the µi , · µi , if the following conditions are satisﬁed: written µ = i∈I

(1) µ =

∗

µi .

i∈I

(2) Every µi is a normal fuzzy subgroup of µ. ∗ (3) µj ∩( µi ) = µ(e){e} ∀ j ∈ I. i∈I\{j}

In addition, when I = {1, 2, . . . , n} (n ∈ N\{1}), we sometimes write · · · · µ1 ⊗ µ2 ⊗ . . . ⊗ µn for µi and call it the direct product of the µi ’s. i∈I

Theorem 1.5.9. Let µ ∈ F(G) and µi ∈ F(G) for all i ∈ I. Suppose that µi (e) = µj (e) ∀i, j ∈ I. Then a necessary and suﬃcient condition for µ = · µi is that i∈I

µ∗ =

·

µ∗i and µ =

i∈I

∗

µi .

i∈I

Proof. The suﬃciency follows directly from Theorem 1.5.7 with Hi = µ∗i and H = µ∗ . · For the necessity, suppose that µ = µi . Then by Theorem 1.5.4, we ∗

obtain that µ =

i∈I

∗ i∈I

µ∗i

and by Theorem 1.4.4, it follows that for all j ∈ I, µ∗j

is a normal subgroup of µ∗ . Again, by Theorem 1.5.4, x ∈ µ∗j ∩ (

∗

µ∗i ) ⇔ x ∈ µ∗j ∩ (

i∈I\{j}

∗

µi )∗

i∈I\{j}

⇔ µj (x) > 0 and ( ⇔ (µj ∩ (

∗

∗

µi )(x) > 0

i∈I\{j}

µi ))(x) > 0

i∈I\{j}

⇔ x = e. for all j ∈ I. Therefore, the desired result holds.

Theorem 1.5.10. Let µ ∈ F(G) and {µi | i ∈ I} be a collection of fuzzy subgroups of G such that µi ⊆ µ ∀ i ∈ I. Suppose that either

1.5 Complete and Weak Direct Products

27

(1) ∪i∈I |µi (G)| is ﬁnite or · µ∗i . (2) µ∗ = i∈I

Then µ =

∗

µi if and only if µa =

i∈I\{j}

∗

(µi )a ∀ a ∈ {b ∈ [0, 1] |

i∈I\{j}

b ≤ µ(e)}. Proof. We ﬁrst show that µi (e) = µ(e) ∀ i ∈ I in either direction of the proof. ∗ Suppose that µ = µi . Then µ(e) = ∨{∧{µi (xi ) | i ∈ I} | e = i∈I xi } ≤ i∈I ∨{∧{µi (e) | i ∈ I} | e = i∈I xi } (since µi (xi ) ≤ µ(e)) = ∧{µi (e) | i ∈ I} ∗ ≤ µ(e) (since µi ⊆ µ). Thus µ(e) = µi (e) ∀i ∈ I. Suppose µa = (µi )a ∀ ∗

a ∈ {b ∈ [0, 1] | b ≤ µ(e)}. Now e ∈ µµ(e) and so e ∈

i∈I

(µi )µ(e) . Thus (µi )µ(e)

i∈I

= ∅. Hence µi (e) ≥ µ(e) ∀i ∈ I. ∗ Suppose that µ = µi . Then i∈I

x∈(

∗

µi )a ⇔ (

∗

µi )(x) ≥ a

⇔ ∨{∧{µi (xi ) | i ∈ I} | x = i∈I xi } ≥ a xi and ∀i ∈ I ⇐ µi (xi ) ≥ a for some representation of x =

i∈I

i∈I

⇔x∈

∗

i∈I

(µi )a .

i∈I

If condition (2) holds, then x has a unique representation. Clearly then, either condition (1) or condition (2) implies that the ⇐ becomes ⇔ in the above ∗ sequence of implications. Conversely, suppose that µa = (µi )a ∀ a ∈ {b ∈ i∈I

[0, 1] | b ≤ µ(e)}. Let x ∈ G and µ(x) = a. Then x ∈ µa and so there exist yi ∈ (µi )a such that x = yi . Thus i∈I ∗ ( µi )(x) = ∨{∧{µi (xi ) | i ∈ I} | x = xi } ≥ a = µ(x). i∈I

i∈I

Hence it follows that µ =

∗

µi .

i∈I

Theorem 1.5.11. Let µ ∈ F(G) and {µi | i ∈ I} be a collection of fuzzy subgroups of G such that µi ⊆ µ ∀ i ∈ I. Suppose that µi (e) = µj (e) ∀i, j ∈ I. Then · · µ= µi if and only if µa = (µi )a i∈I

∀ a ∈ {b ∈ [0, 1] | b ≤ µ(e)}\{0}.

i∈I

28

1 Fuzzy Subsets and Fuzzy Subgroups

Proof. Suppose that µ =

·

µi . Then µ∗ =

i∈I

a ∈ {b ∈ [0, 1] | b ≤ µ(e)}\{0}, µa =

∗

· i∈I

µ∗i by Theorem 1.5.9. Thus ∀

(µi )a by Theorem 1.5.10 and

i∈I

(µi )a ∩ (µj )a ⊆ µ∗i ∩ µ∗j = {e} if i = j. Thus µa =

· (µi )a i∈I

∀ a ∈ {b ∈ [0, 1] | b ≤ µ(e)}\{0} by Theorem 1.4.3. Conversely, suppose that · µa = (µi )a ∀ a ∈ {b ∈ [0, 1] | b ≤ µ(e)}\{0}. Then by Theorem 1.5.10, µ ∗

=

i∈I

i∈I

µi . Let x ∈ µ∗i ∩ µ∗j . Then for a = µi (x) ∧ µj (x), we have that a > 0

and x ∈ (µi )a ∩ (µj )a = {e} for i = j. Hence x = e and so µ∗i ∩ µ∗j = {e} for · · i = j. Thus µ∗ = µ∗i by Theorem 1.4.4 and hence µ = µi by Theorem i∈I

i∈I

1.5.9.

Theorem 1.5.12. Let µ ∈ F(G) and {µi | i ∈ I} be a collection of fuzzy · subgroups of G such that µi ⊆ µ ∀ i ∈ I. Suppose that µ = µi and µ∗ = ∗

(µi )∗ . Then µ∗ =

i∈I

·

i∈I

(µi )∗ .

i∈I

Proof. By Corollary 1.2.7, µ∗ is a subgroup of G. Since by deﬁnition, every µi is a normal fuzzy subgroup of µ, it follows from Theorem 1.4.4 that every ∗ (µi )∗ is a normal subgroup of µ∗ . Let j ∈ I and x ∈ (µj )∗ ∩ (µi )∗ . Then x∈

µ∗j ∩

µ∗ =

·

∗

µ∗i

i∈I\{j}

and so (µj ∩ (

i∈I\{j}

∗

µi ))(x) > 0. Hence x = e. Therefore,

i∈I\{j}

(µi )∗ .

i∈I

The following two theorems are direct consequences of Theorems 1.4.5 and 1.4.6, respectively. Theorem 1.5.13. Let µ ∈ F(G), {µi | i ∈ I} be a collection of fuzzy subgroups of G such that µi ⊆ µ ∀ i ∈ I. Let νi ∈ N F(G). Suppose that µ = · ∗ · µi . Let ξ = (µi ∩ νi ). Then ξ = (µi ∩ νi ). i∈I

i∈I

i∈I

Theorem 1.5.14. Let µ ∈ F(G) and {µi | i ∈ I} be a collection of fuzzy subgroups of G such that µi ⊆ µ ∀ i ∈ I. Suppose that for all i ∈ I, νi is

1.5 Complete and Weak Direct Products ·

a normal fuzzy subgroup of µ and µ = ·

ξ=

µi . Let ξ =

i∈I

∗

29

(µi ∩ νi ). Then

i∈I

(µi ∩ νi ).

i∈I

We now consider some results from [3] and [12]. Let I = {1, 2, ..., n}. Let Gi ∼

∼

∼

be a group, i = 1, 2, ..., n, and let µ a fuzzy subgroup of G1 ⊗ G2 ⊗ ... ⊗ Gn . We show there exist fuzzy subgroups µ1 , µ2 , ..., µn of G1 , G2 , ..., Gn , respectively, ∼ such that the complete direct product µi ⊆ µ. We ﬁnd necessary and suﬃcient conditions for µ =

∼

i∈I

µi .

i∈I

For all i = 1, 2, ..., n, let µi be a fuzzy subgroup of a group Gi . Let ei denote the identity of Gi , i = 1, 2, ..., n. Recall that the complete direct product of ∼ ∼ ∼ ∼ the µi (i = 1, 2, ..., n) is the fuzzy subset µi of G1 ⊗ G2 ⊗ ... ⊗ Gn deﬁned by (

∼

i∈I

µi )(x1 , x2 , ..., xn ) = µ1 (x1 ) ∧ µ2 (x2 ) ∧ ... ∧ µn (xn ) for all xi ∈ Gi , i =

i∈I

1, 2, ..., n. Theorem 1.5.15. Let G1 , G2 , .., Gn be groups and let µ be a fuzzy subgroup ∼ ∼ ∼ of G1 ⊗ G2 ⊗ ... ⊗ Gn . Let µi be the fuzzy subset of Gi deﬁned by µi (x) = µ(e1 , ..., ei−1 , x, ei+1 , ..., en )∀x ∈ Gi , i = 1, 2, ..., n. Then µi is a fuzzy subgroup ∼ of Gi , i = 1, 2, ..., n, and µi ⊆ µ. i∈I ∼

∼

∼

∼

∼

∼

Proof. Let Ki = {e1 } ⊗ ... ⊗ {ei−1 } ⊗ Gi ⊗ {ei+1 } ⊗ ... ⊗ {en } for i = ∼

∼

∼

1, 2, ..., n. Then Ki is a subgroup of G1 ⊗ G2 ⊗ ... ⊗ Gn . Clearly, µ|Ki is a fuzzy subgroup of Ki . Let φ : Gi → Ki be the function deﬁned by φ(x) = (e1 , ..., ei−1 , x, ei+1 , ..., en )∀x ∈ Gi . Then φ is an isomorphism and µi = µ|Ki ◦ φ. Hence µi is a fuzzy subgroup of Gi . Now µ(x1 , x2 , ..., xn ) = µ((x1 , e2 , ..., en )(e1 , x2 , ..., xn )) ≥ µ(x1 , e2 , ..., en ) ∧ µ(e1 , x2 , ..., xn ) = µ1 (x1 ) ∧ µ(e1 , x2 , x3 , ..., xn ) = µ1 (x1 ) ∧ ((µe1 , x2 , e3 , ..., en )(e1 , e2 , x3 , ..., xn )) ≥ µ1 (x1 ) ∧ µ((e1 , x2 , e3 , ..., en ) ∧ µ(e1 , e2 , x3 , ..., xn )) = µ1 (x1 ) ∧ (µ2 (x2 ) ∧ µ(e1 , e2 , x3 , ..., xn )) ≥ µ1 (x1 ) ∧ (µ2 (x2 ) ∧ (µ3 (x3 ) ∧ µ(e1 , e2 , e3 , x4 , ..., xn ))) ≥ ... ≥ µ1 (x1 ) ∧ (µ2 (x2 ) ∧ ... ∧ (µn−2 (xn−2 ) ∧ (µn−1 (xn−1 ) ∧ µn (xn ))) = µ1 (x1 ) ∧ µ2 (x2 ) ∧ ... ∧ µn−2 (xn−2 ) ∧ µn−1 (xn−1 ) ∧ µn (xn ). ∼ Thus µ(x1 , x2 , x3 , ..., xn ) ≥ µ1 (x1 )∧µ2 (x2 )∧...∧µn (xn ) = ( µi )(x1 , x2 , ..., xn ). i∈I

30

1 Fuzzy Subsets and Fuzzy Subgroups

Lemma 1.5.16. Let G1 , G2 , ..., Gn be groups and let µ be a fuzzy subgroup ∼ ∼ ∼ of G1 ⊗ G2 ⊗ ... ⊗ Gn . If µ(e1 , e2 , ..., ei−1 , xi , ei+1 , ..., en ) ≥ µ(x1 , x2 , ..., xn ) for i = 1, 2, ..., k − 1, k + 1, ..., n, then µ(e1 , e2 , ..., ek−1 , xk , ek+1 , ..., en ) ≥ µ(x1 , x2 , ..., xn ). Proof. µ (e1 , ..., ek−1 , xk , ek+1 , ..., en ) = −1 −1 −1 µ((x1 , x2 , ..., xn )(x−1 1 , ..., xk−1 , ek , xk+1 , ..., xn )) ≥ −1 −1 −1 −1 µ(x1 , x2 , ..., xn ) ∧ (µ(x1 , x2 , ..., xk−1 , ek , x−1 k+1 , ..., xn )) = µ(x1 , x2 , ..., xn ) ∧ µ(x1 , x2 , ..., xk−1 , ek , xk+1 , ..., xn ) ≥ µ(x1 , x2 , ..., xn ) ∧ (µ(e1 , e2 , ..., ek−2 , xk−1 , ek , ek+1 , ..., en )∧ µ(x1 , x2 , ..., xk−2 , ek−1 , ek , xk+1 , ..., xn )) = µ(x1 , x2 , ..., xn ) ∧ µ(x1 , ..., xk−2 , ek−1 , ek , xk+1 , ..., xn ) ≥ µ(x1 , x2 , ..., xn ) ∧ (µ(e1 , ..., ek−3 , xk−2 , ek−1 , ek , ek+1 , ..., en )∧ µ(x1 , ..., xk−3 , ek−2 , ek−1 , ek , xk+1 , ..., xn )) = µ(x1 , x2 , ..., xn ) ∧ µ(x1 , ..., xk−3 , ek−2 , ek−1 , ek , xk+1 , ..., xn ) ≥ ... ≥ µ(x1 , x2 , ..., xn ) ∧ µ(e1 , ..., ek−2 , ek−1 , ek , ek+1 , ..., en−1 , xn ) = µ(x1 , x2 , ..., xn ). Thus µ(e1 , e2 , ..., ek−1 , xk , ek+1 , ..., en ) ≥ µ(x1 , x2 , ..., xn ).

Theorem 1.5.17. Let G1 , G2 , ..., Gn be groups and let µ be a fuzzy subgroup ∼ ∼ ∼ of G1 ⊗ G2 ⊗ ... ⊗ Gn . Then for all xi ∈ Gi , µ(e1 , e2 , ..., ei−1 , xi , ei+1 , ..., en ) ≥ ∼ µ(x1 , x2 , ..., xn ) for i = 1, 2, ..., k − 1, k + 1, ..., n, if and only if µ = µi , i∈I

where µ1 , µ2 , ..., µn are fuzzy subgroups of G1 , G2 , ..., Gn , respectively. Proof. Deﬁne µj : Gj → [0, 1] by µj (xj ) = µ(e1 , ..., xj , ..., en ) for j = 1, 2, ..., n. ∼

∼

∼

∼

∼

Let Kj = {e1 } ⊗ {e2 } ⊗ ... ⊗ Gj ⊗ ... ⊗ {en } . Then each µ|Kj is a fuzzy sub∼ µi ⊆ µ. By Lemma group of Kj for j = 1, 2, ..., n. By Theorem 1.5.15, i∈I

1.5.16, µ(e1 , e2 , ..., ei−1 , xi , ei+1 , ..., en ) ≥ µ(x1 , x2 , ..., xn ) for i = 1, 2, ..., n. µj (xj ) = µ(e1 , ..., xj , ..., en ) ≥ µ(x1 , x2 , ..., xn ) for j = 1, 2, ..., n. Thus ∼ ( µi )(x1 , x2 , ..., xn ) = µ1 (x1 )∧µ2 (x2 )∧...∧µn (xn ) ≥ µ(x1 , x2 , ..., xn ). Hence i∈I

∼

µi = µ. The other direction is straightforward.

i∈I

Lemma 1.5.18. Let G be a group and let µ be a fuzzy subgroup of G. Let x ∈ G be of ﬁnite order k. If r ∈ N and k are relatively prime, then µ(x) = µ(xr ). Proof. There exist integers s and t such that 1 = sr + kt. Thus µ(x) = µ(xsr+tk ) = µ(xsr xtk ) = µ(xsr ) ≥ µ(xr ) ≥ µ(x). Lemma 1.5.19. Let G1 , G2 , ..., Gn be ﬁnite groups and let µ be a fuzzy sub∼ ∼ ∼ group of G1 ⊗ G2 ⊗ ... ⊗ Gn . If the orders of Gi and Gj are relatively prime for all i, j ∈ I, i = j, then µ(x1 , ..., xi , ..., xn ) ≤ µ(e1 , ....ei−1 , xi , ei+1 , ..., en ) for all xi ∈ Gi , i = 1, 2, ..., n.

1.6 Fuzzy Order Relative to Fuzzy Subgroups ∼

∼

31

∼

Proof. Let (x1 , ..., xi , ..., xn ) ∈ G1 ⊗ G2 ⊗ ... ⊗ Gn . Let i ∈ I and ri = |G1 |...|Gi−1 ||Gi+1 |...|Gn |. Then ri and |Gi | are relatively prime as are ri and the order of xi for all i ∈ I. Clearly, (xj )ri = ej for all i, j ∈ I, j = i. Hence µ(x1 , ..., xi , ..., xn ) ≤ µ((x1 , ..., xi , ..., xn )ri ) = µ((x1 , ei , ..., en )ri ...(e1 , ..., ei−1 , xi , ei+1 , ..., en )ri ...(e1 , ..., en−1 , xn )ri ) = µ(e1 , ..., ei−1 , xri i , ei+1 , ..., en ) = µ(e1 , ..., ei−1 , xi , ei+1 , ..., en ), where the last equality holds by Lemma 1.5.18. Theorem 1.5.20. Let G1 , G2 , ..., Gn be ﬁnite groups and let µ be a fuzzy sub∼ ∼ ∼ group of G1 ⊗ G2 ⊗ ... ⊗ Gn . If the orders of Gi and Gj are relatively prime ∼ µi , where µ1 , µ2 , ..., µn are fuzzy subgroups for all i, j ∈ I, i = j, then µ = i∈I

of G1 , G2 , ..., Gn , respectively. Proof. The result follows from Theorem 1.5.17 and Lemma 1.5.19.

1.6 Fuzzy Order Relative to Fuzzy Subgroups We present the notion of the fuzzy order of an element of a group relative to a fuzzy subgroup as introduced by Kim [26]. This notion generalizes the usual order of an element. We show that most of the basic properties of the order of an element in standard group theory are valid in the theory of fuzzy subgroups if the order of an element is replaced by fuzzy order. We denote the identity element of a group G by e, the order of an element x of G by o(x), and the greatest common divisor of integers m and n by (m, n). Proposition 1.6.1. Let G be a ﬁnite group and let µ be a fuzzy subgroup of G. If o(y)|o(x) and x, y ∈ z for some z ∈ G, then µ(x) ≤ µ(y). Proof. Let o(y) = k. Then o(x) = kq for some q ∈ N. Now y = z i and x = z j for some i, j ∈ Z. Hence z ik = e = z jkq . Thus y = xq . Hence µ(y) = µ(xq ) ≥ µ(x). We now recall some basic results concerning the orders of elements of a group. Theorem 1.6.2. Let G be a group and let x, y, z ∈ G. Then the following assertions hold. (1) If xm = e, then o(x)| m, where m ∈ Z. (2) o(xm ) = o(x)/(o(x), m), where m ∈ Z. (3) If (o(x), o(y)) = 1 and xy = yx, then o(xy) = o(x) · o(y). (4) If z = y −1 xy, then o(z) = o(x). (5) If o(z) = mn with (m, n) = 1, then z = xy = yx for some x, y ∈ G with o(x) = m and o(y) = n. Furthermore, such an expression for z is unique.

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We next deﬁne the fuzzy order of an element of a group and investigate some of its properties. Speciﬁcally, we show that Theorem 1.6.2 is valid if the orders of elements are replaced by their fuzzy orders. Deﬁnition 1.6.3. Let µ be a fuzzy subgroup of a group G and let x ∈ G. If there exists a positive integer n such that µ(xn ) = µ(e), then the least such positive integer is called the fuzzy order of x with respect to µ and written as F Oµ (x). If no such n exists, x is said to be of inﬁnite fuzzy order with respect to µ. Equality of o(x) and o(y) does not imply that of F Oµ (x) and F Oµ (y), as is shown in the following example. Example 1.6.4. Let G = {e, a, b, ab} be the Klein four-group. Deﬁne the fuzzy subgroup µ of G by µ(e) = µ(ab) = t0 and µ(a) = µ(b) = t1 , where t0 > t1 . Clearly, o(a) = o(ab) = 2, but F Oµ (a) = 2 and F Oµ (ab) = 1. Also observe that in this example, o(a)|o(ab), but µ(ab) > µ(a). Thus Proposition 1.6.1 doesn’t hold here since the elements a and ab do not lie in the same cyclic subgroup of G. Proposition 1.6.5. Let µ be a fuzzy subgroup of a group G. For x ∈ G, if µ(xm ) = µ(e) for some integer m, then F Oµ (x)|m. Proof. Let F Oµ (x) = n. By the Euclidean algorithm, there exist integers s and t such that m = ns + t, where 0 ≤ t < n. Then µ(xt ) = µ(xm−ns ) = µ(xm (xn )−s ) ≥ µ(xm ) ∧ µ((xn )−s ) ≥ µ(e) ∧ µ(xn ) = µ(e) ∧ µ(e) = µ(e). Thus µ(xt ) = µ(e). Hence t = 0 by the minimality of n.

If o(x) is ﬁnite, then F Oµ (x) is clearly ﬁnite for all fuzzy subgroups µ of G. If o(x) is inﬁnite, then for each positive integer n, there exists a fuzzy subgroup µn of G such that F Oµn (x) = n as the following example shows. Example 1.6.6. Let x be an element of inﬁnite order in the group G. For each positive integer n, deﬁne the fuzzy subgroup µn of G by t0 if y ∈ xn , µn (y) = t1 otherwise, where t0 > t1 . Observe that F Oµn (x) = n. With this example in mind, the following corollary assumes added importance when one agrees that in the extended real number system ∞ is divisible by both ∞ and a positive real number.

1.6 Fuzzy Order Relative to Fuzzy Subgroups

33

Corollary 1.6.7. Let µ be a fuzzy subgroup of a group G. Then F Oµ (x)|o(x) for all x ∈ G. Proposition 1.6.8. Let µ be a fuzzy subgroup of a group G and let x and y be elements of G such that (F Oµ (x), F Oµ (y)) = 1 and xy = yx. If µ(xy) = µ(e), then µ(x) = µ(y) = µ(e). Proof. Let F Oµ (x) = n and F Oµ (y) = m. Then µ(e) = µ(xy) ≤ µ((xy)m ) = µ(xm y m ). Hence µ(xm y m ) = µ(e). Now µ(xm ) = µ(xm y m y −m ) ≥ µ(xm y m ) ∧ µ(y −m ) = µ(e) ∧ µ(e) = µ(e). Thus µ(xm ) = µ(y m ) = µ(e). Therefore, n|m by Proposition 1.6.5. But (n, m) = 1. Thus n = 1, i.e., µ(x) = µ(e). Similarly µ(y) = µ(e). Proposition 1.6.8 is an improvement of the result that µ(xy −1 ) = µ(e) implies that µ(x) = µ(y). By virtue of Corollary 1.6.7, the assumption of the proposition can be weakened as follows: Corollary 1.6.9. Let µ be a fuzzy subgroup of a group G, and let x and y be elements of G such that (o(x), o(y)) = 1 and xy = yx. If µ(xy) = µ(e), then µ(x) = µ(y) = µ(e). Neither the assumption (F Oµ (x), F Oµ (y)) = 1 in Proposition 1.6.8 nor the assumption (o(x), o(y)) = 1 in Corollary 1.6.9 can be omitted. In fact, in Example 1.6.4, µ(a) = µ(b) = µ(e) and µ(ab) = µ(e), but F Oµ (a) = F Oµ (b) = o(a) = o(b) = 2. Theorem 1.6.10. Let µ be a fuzzy subgroup of a group G. Let F Oµ (x) = n, where x ∈ G. If m is an integer with d = (m, n), then F Oµ (xm ) = n/d. Proof. Let F Oµ (xm ) = t. First, we have n/d

µ((xm )

) = µ(xnk ) for some integer k ≥ µ(xn ) = µ(e).

Thus t|n/d by Proposition 1.6.5. Also, since d = (m, n), there exist integers i and j such that ni + mj = d. We then have µ(xtd ) = µ(xt(ni+mj) ) = µ(xnti xmtj )

j ti t ≥ µ((xn ) ) ∧ µ( (xm ) ) ≥ µ(xn ) ∧ µ((xm )t ) = µ(e) ∧ µ(e) = µ(e). This implies that n|td by Proposition 1.6.5, i.e., n/d|t. Consequently, t = n/d. Proposition 1.6.11. Let µ be a fuzzy subgroup of a group G. Let F Oµ (x) = n where x ∈ G. If m is an integer with (n, m) = 1, then µ(xm ) = µ(x).

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Proof. Since (n, m) = 1, there exist integers s and t such that ns + mt = 1. We then have µ(x) = µ(xns+mt ) = µ((xn )s (xm )t ) ≥ µ((xn )s ) ∧ µ((xm )t ) ≥ µ(xn ) ∧ µ(xm ) = µ(e) ∧ µ(xm ) = µ(xm ) ≥ µ(x). Proposition 1.6.11 is an improvement of Lemma 1.5.18. Theorem 1.6.12. Let µ be a fuzzy subgroup of a group G. Let F Oµ (x) = n, where x ∈ G. If i ≡ j(modn), where i, j ∈ Z, then F Oµ (xi ) = F Oµ (xj ). Proof. Let F Oµ (xi ) = t and F Oµ (xj ) = s. By the assumption, i = j + nk for some integer k. We then have ks

µ((xi )s ) = µ((xj+nk )s ) = µ((xj )s (xn ) ) ks

≥ µ((xj )s ) ∧ µ((xn ) ) ≥ µ(e) ∧ µ(xn ) = µ(e) ∧ µ(e) = µ(e), and so t|s. Similarly, s|t. Thus we have t = s.

Theorem 1.6.13. Let µ be a fuzzy subgroup of a group G, and let x and y be elements of G such that xy = yx and (F Oµ (x), F Oµ (y)) = 1. Then F Oµ (xy) = F Oµ (x) · F Oµ (y). Proof. Let F Oµ (xy) = n, F Oµ (x) = s, and F Oµ (y) = t. Then µ((xy)st ) = µ(xst y st ) ≥ µ((xs )t ) ∧ µ((y t )s ) ≥ µ(xs ) ∧ µ(y t ) = µ(e) ∧ µ(e) = µ(e). Thus n|st by Proposition 1.6.5. Now µ (e) = µ((xy)n ) = µ(xn y n ). Since n|st, n|s or n|t, say n|t. Then n and s are relatively prime. Hence F Oµ (xn ) = s, by Proposition 1.6.11. By t . Since s and t are relatively prime, s and (n,t t) Theorem 1.6.10, µ(y n ) = (n,t) are relatively prime. Thus (F Oµ (xn ), F Oµ (y n )) = 1. Therefore, µ(xn ) = µ(y n ) = µ(e) by Proposition 1.6.8, and so both s and t divide n by Proposition 1.6.5. Therefore, st|n since (s, t) = 1. Hence we have n = st. By virtue of Corollary 1.6.7, the assumption of Theorem 1.6.13 can be weakened as follows: Corollary 1.6.14. Let µ be a fuzzy subgroup of a group G, and let x and y be elements of G such that xy = yx and (o(x), o(y)) = 1. Then F Oµ (xy) = F Oµ (x) · F Oµ (y).

1.6 Fuzzy Order Relative to Fuzzy Subgroups

35

In Theorem 1.6.13, even if µ were normal, the assumption xy = yx could not be omitted as can be seen by the following example. Example 1.6.15. Deﬁne the fuzzy subset µ of the symmetric group S4 of degree 4 by t0 if x = e, µ(x) = t1 otherwise, where t0 > t1 . Then µ is a normal fuzzy subgroup of S4 . Now, let x = (1 2) and y = (2 3 4). Then F Oµ (x) = 2, F Oµ (y) = 3, F Oµ (xy) = F Oµ (yx) = 4, and xy = yx. Theorem 1.6.16. Let µ be a fuzzy subgroup of a group G. For z ∈ G, if F Oµ (z) = mn with (m, n) = 1, then there exist x and y in G such that z = xy = yx, F Oµ (x) = m, and F Oµ (y) = n. Furthermore, such an expression for z is unique in the sense of fuzzy grades, i.e., if (x, y) and (x1 , y1 ) are such pairs, then µ(x) = µ(x1 ) and µ(y) = µ(y1 ). Proof. Since (m, n) = 1, there exist integers s and t such that ms + nt = 1. Here (m, t) = (n, s) = 1. Let x = z nt and y = z ms . Then z = xy = yx and by Theorem 1.6.10, F Oµ (x) = F Oµ (z nt ) = m and F Oµ (y) = F Oµ (z ms ) = n. This proves the existence of x and y. For the uniqueness, let (x, y) and (x1 , y1 ) be pairs satisfying the assumption. Then, since F Oµ (x) = F Oµ (x1 ) = m and F Oµ (y) = F Oµ (y1 ) = n, we obtain µ(x) = µ(x1−ms ) = µ(xnt ) = µ(xnt y nt ) = µ((xy)nt ) nt nt = µ((x1 y1 )nt ) = µ(xnt 1 y1 ) = µ(x1 ) = µ(x1−ms ) = µ(x1 ). 1 Similarly, µ(y) = µ(y1 ). This proves the uniqueness of (x, y).

Theorem 1.6.17. Let µ be a normal fuzzy subgroup of a group G. Then F Oµ (x) = F Oµ (y −1 xy) for all x, y ∈ G. Proof. Let x, y ∈ G. Then we have µ(xn ) = µ(y −1 xn y) = µ((y −1 xy)n ) for all n ∈ Z. Thus F Oµ (x) = F Oµ (y −1 xy). The next example shows that Theorem 1.6.17 is not valid if µ is not normal in G. The notation a, b|a3 = b2 = e, ba = a2 b denotes the group generated by a and b, where a and b satisfy the indicated conditions. It is common to refer to this notation as a presentation of the group. Example 1.6.18. Let D3 = a, b|a3 = b2 = e, ba = a2 b be the dihedral group with six elements. Deﬁne a fuzzy subgroup µ of D3 by t0 if x ∈ b , µ(x) = t1 otherwise, where t0 > t1 . Then a−1 ba ∈ / b , and so F Oµ (b) = 1 = F Oµ (a−1 ba).

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1.7 Fuzzy Orders in Cyclic Groups Lemma 1.7.1. Let µ be a fuzzy subgroup of a cyclic group G and let a and b be any two generators of G. Then F Oµ (a) = F Oµ (b). Proof. Suppose G is a ﬁnite cyclic group with |G| = n. Since G is generated by a and b, we get o(a) = o(b) = n. Further b = am for some m ∈ Z and so we must have (m, n) = 1. Therefore, F Oµ (a) = F Oµ (am ) = F Oµ (b) = n by Theorem 1.6.10. If G is an inﬁnite group, then b = a−1 . Theorem 1.7.2. Let µ be a fuzzy subgroup of a cyclic group G of ﬁnite order n. Then the following assertions hold for all x, y ∈ G. (1) If o(x) = o(y), then F Oµ (x) = F Oµ (y). (2) If o(x)|o(y), then F Oµ (x)|F Oµ (y). (3) If o(x) > o(y), then F Oµ (x) ≥ F Oµ (y). Proof. Let G = a . Let x = as , y = at , and F Oµ (a) = m. By Lemma 1.7.1, m is independent of a particular choice of a generator a of G. Thus o(x) = n/(s, n), o(y) = n/(t, n), F Oµ (x) = m/(s, m), F Oµ (y) = m/(t, m), and m|n by Corollary 1.6.7 and Theorem 1.6.10. (1) The result here follows from (2). (2) If o(x)|o(y), then (t, n)|(s, n), and so (t, m)|(s, m) since m|n. Thus F Oµ (x)|F Oµ (y). (3) If o(x) > o(y), then (s, n) < (t, n), and so (s, m) ≤ (t, m) since m|n. Hence F Oµ (x) ≥ F Oµ (y). Theorem 1.7.3. Let µ be a fuzzy subgroup of a cyclic group G of ﬁnite order. Then the following assertions hold for all x, y ∈ G. (1) If F Oµ (x) = F Oµ (y), then µ(x) = µ(y). (2) If F Oµ (x)|F Oµ (y), then µ(x) ≥ µ(y). Proof. Let G = a . Let x = as , y = at , and F Oµ (a) = m. By Lemma 1.7.1, m is independent of a particular choice of a generator a of G. Then F Oµ (x) = m/(s, m) and F Oµ (y) = m/(t, m), by Theorem 1.6.10. Let s = h(s, m), t = i(t, m), and m = j(t, m) = k(s, m) for some h, i, j, k ∈ Z. If F Oµ (x)|F Oµ (y), then (t, m)|(s, m). Thus t divides si = h(s, m)i and m divides sj = h(s, m)j, and hence we have µ(x) = µ(as ) = µ(as(iv+jw) ) for some u, w ∈ Z since (i, j) = 1 = µ(asiv asjw ) ≥ µ(asiv ) ∧ µ(asjw ) ≥ µ(at ) ∧ µ(am ) = µ(y) ∧ µ(e) = µ(y). This proves (2). Condition (1) follows from (2).

Here is a counterexample to show that part (2) of Theorem 1.7.3 may fail to hold if G is not cyclic:

References

37

Example 1.7.4. Let H be a cycle group of order 4 generated by x and let K be a cyclic group of order 2 generated by y. Let G be the direct product of H and K, G = {(u, v)|u ∈ H, v ∈ K}. Deﬁne the fuzzy subset µ of G as follows: µ(e, e) = 1, µ(x2 , e) = 3/4, µ(x, e) = µ(x3 , e) = 1/2 and µ(u, v) = 1/4 for all other (u, v). Then the level subsets of µ are subgroups of G, and so µ is a fuzzy subgroup of G. Now F Oµ (x, e) = 4 > 2 = F Oµ (e, y), but µ(x, e) = 1/2 > 1/4 = µ(e, y). Thus F Oµ (e, y)|F Oµ (e, x), but µ(y, e) = 1/4 < 1/2 = µ(e, x). Theorem 1.7.3 is an improvement on Proposition 1.6.1. However, by combining Theorem 1.7.2 with Theorem 1.7.3, we have Proposition 1.6.1. Corollary 1.7.5. Let µ be a fuzzy subgroup of a cyclic group G of ﬁnite order. Then the following assertions hold for all x, y ∈ G. (1) If o(x) = o(y), then µ(x) = µ(y). (2) If o(x)|o(y), then µ(x) ≥ µ(y). Remark 1.7.6. Observe that Example 1.6.4 shows that the conclusions of Theorems 1.7.2 and Corollary 1.7.5 may fail to hold in ﬁnite non-cyclic groups. Remark 1.7.7. A natural question that comes to mind after reading results from Section 1.7 is the following: Let µ be a fuzzy subgroup of a ﬁnite cyclic group G and let x, y be any two elements of G. If F Oµ (x) ≥ F Oµ (y), then does it follow that µ(y) ≥ µ(x)? The answer is “no” as can be seen by the following example: Example 1.7.8. Let G = {e, a, a2 , a3 , . . . , a6 } be a cyclic group of order 6 generated by an element a. Deﬁne the fuzzy subgroup µ of G by µ(e) = 1, µ(a2 ) = µ(a4 ) = 3/4, µ(z) = 1/4 for all other z ∈ G. Then one can see that µ is a fuzzy subgroup of G. Now F Oµ (a2 ) = 3 and F Oµ (a3 ) = 2. But µ(a2 ) = 3/4 and µ(a3 ) = 1/4. Open Question • Characterize groups G which satisfy conditions (1) and (2) of Theorem 1.7.3.

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5. J. M. Anthony and H. Sherwood, A characterization of fuzzy subgroups, J. Math. Anal. Appl. 69 (1979) 297-305. 6. M. Asaad, Groups and fuzzy subgroups, Fuzzy Sets and Systems 39 (1991) 323328. 7. S. K. Bhakat and P. Das, A note on fuzzy Archimedean ordering, Fuzzy Sets and Systems 91 (1997) 91-94. 8. K. R. Bhutani, Fuzzy Sets, Fuzzy Relations and Fuzzy Groups: Some Interrelations, Inform. Sci. 73(1993), 107-115. 9. R. Biswas, Fuzzy subgroups and antifuzzy subgroups. Fuzzy Sets and Systems 35 (1990) 121-124. 10. D-G Chen and W-X Gu, Product structure of the fuzzy factor groups, Fuzzy Sets and Systems 60 (1993) 229-232. 11. D-G Chen and W-X Gu, Generating fuzzy factor groups and fundamental theorem of isomorphism, Fuzzy Sets and Systems 82 (1996) 357-360. 12. I. Chon, Fuzzy subgroups as products, Fuzzy Sets and Systems, to appear. 29 13. P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264-269. 1 14. V. N. Dixit, S. K. Bhambri, and P. Kumar, Union of fuzzy subgroups, Fuzzy Sets and Systems 78 (1996) 121-123. 15. V. N. Dixit, R. Kumar, and N. Ajamal, Level subgroups and union of fuzzy subgroups, Fuzzy Sets and Systems 37 (1990)359-371. 16. M. S. Eroglu, The homomorphic image of a fuzzy subgroup is always a fuzzy subgroup, Fuzzy Sets and Systems 33 (1989) 255-256. 17. L. Filep, Structure and construction of fuzzy subgroups of a group, Fuzzy Sets and Systems 51 (1992) 105-109. 18. L. Filep and I. Gy. Maurer, Compatible fuzzy relations and groups, Studia Scientiarum Mathematicarum Hungarica 24 (1989) 345-348. 19. L. Fuchs, Inﬁnite Abelian Groups I, Pure and Applied Math., Vol. 36, Academic Press, New York 1970. 20. W-X Gu and D-G Chen, LP-fuzzy groups, J. Fuzzy Math. 1 (1993) 343-357. 21. W-X Gu and D-G Chen, A fuzzy subgroupoid which is not a fuzzy group, Fuzzy Sets and Systems 62 (1994) 115-116. 22. K. C. Gupta and S. Ray, Protogroups generated by fuzzy sets, Inf. Sci. 73 (1993) 1-16. 23. K. C. Gupta and S. Ray, The Schnitt axiom and unions of fuzzy subgroups, Inf. Sci. 70 (1993) 213-220. 24. A. K. Ibrahim and S. A. Khatab, Pruc. Pakistan Acad. Sci. 27 (1990) 46-53. 25. A. Jain and N. Ajmaal, A new approach to the theory of fuzzy groups, J. Fuzzy Math. 12 (2004) 341-355. 26. J. G. Kim, Fuzzy orders relative to fuzzy subgroups, Inform. Sci. 80 (1994) 341-348. 31 27. I. J. Kumar, P. K. Saxena, and P. Yadav, Fuzzy normal subgroups and fuzzy quotients, Fuzzy Sets and Systems 46 (1992) 121-132. 28. S, Kundu, The correct form of a recent result on level-subgroups of a fuzzy group, Fuzzy Sets and Systems 97 (1998) 261-263. 29. X. Li and G. Wang, The SH -interval-valued fuzzy group, Fuzzy Sets and Systems 112 (2000) 319-325. 30. W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982)133-139. 1

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2 Fuzzy Caley’s Theorem and Fuzzy Lagrange’s Theorem

We begin our discussion with properties of normal fuzzy subgroups. Fuzzy analogs of some group theoretic concepts such as cosets, characteristic subgroups, conjugate fuzzy subgroups, quotient groups are then presented. We show a number of results using these fuzziﬁed notions. The order of a fuzzy subgroup is discussed as is the notion of a solvable fuzzy subgroup. The fuzziﬁcation of Caley’s Theorem and Lagrange’s Theorem are also presented. In this chapter, we consider primarily, although not exclusively, the work presented in [4, 10, 11].

2.1 Properties of Normal Fuzzy Subgroups In Section 1.3, the notion of normal fuzzy subgroups and some of their properties were discussed. In this section, we continue the discussion of normal fuzzy subgroups and their fuzzy quotient groups. Let S be a groupoid, i.e., a set which is closed under a binary relation denoted multiplicatively. A function µ : S → [0, 1] is called a fuzzy subgroupoid of S if µ(xy) µ(x) ∧ µ(y)∀x, y ∈ S. Let N be a subgroup of a group G. We write N G to denote that N is a normal subgroup of the group G. Lemma 2.1.1. If µ is a fuzzy subgroupoid of a ﬁnite group G, then µ is a fuzzy subgroup. Proof. Let x ∈ G, x = e. Since G is ﬁnite, x has ﬁnite order, say n > 1. Thus xn = e and so x−1 = xn−1 . Now using the deﬁnition of a fuzzy subgroupoid repeatedly, we have that µ(x−1 ) = µ(xn−1 ) = µ(xn−2 x) µ(xn−2 ) ∧ µ(x) ≥ µ(x) ∧ ... ∧ µ(x) = µ(x). Hence µ is a fuzzy subgroup of G. Lemma 2.1.2. Let µ be a fuzzy subgroup of G. Let x ∈ G. Then µ(xy) = µ(y) ∀y ∈ G if and only if µ(x) = µ(e). John Mordeson, Kiran R. Bhutani, Azriel Rosenfeld: Fuzzy Group Theory, StudFuzz 182, 41–60 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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Proof. Suppose that µ(xy) = µ(y) ∀y ∈ G. Then by letting y = e, we get that µ(x) = µ(e). Conversely, suppose that µ(x) = µ(e). Then µ(y) µ(x)∀ y ∈ G and so µ(xy) µ(x) ∧ µ(y) = µ(y). Also, µ(y) = µ(x−1 xy) µ(x) ∧ µ(xy) = µ(xy). Hence µ(xy) = µ(y). Recall that a fuzzy subgroup µ of a group G is called normal if µ(xy) = µ(yx) ∀x, y ∈ G. Recall also from group theory that elements x, y of G are called conjugates if there exists z ∈ G such that x = zyz −1 . The notion of conjugacy is an equivalence relation and the equivalence classes are called conjugacy classes. . Theorem 2.1.3. Let µ be a fuzzy subgroup of a group G. Then µ is normal if and only if µ is constant on the conjugacy classes of G. Proof. Suppose that µ is normal. Then µ(y −1 xy) = µ(xyy −1 ) = µ(x) ∀x, y ∈ G. Conversely, suppose that µ is constant on each conjugacy class of G. Then µ(xy) = µ(xyxx−1 ) = µ(x(yx)x−1 ) = µ(yx) ∀x, y ∈ G. Hence µ is normal. We now give an alternative formulation of the notion of normal fuzzy subgroup in terms of “commutators” of a group. First, we recall that if G is a group and x, y ∈ G, then the element x−1 y −1 xy is usually denoted by [x, y] and is called the commutator of x and y. If x and y commute with each other, then [x, y] = e. If H and K are subgroups of a group G, then [H, K] is deﬁned to be the subgroup generated by {[x, y] | x ∈ H, y ∈ K}. The motivation behind the following theorem is that N G if and only if [N, G] ⊆ N. Theorem 2.1.4. Let µ be a fuzzy subgroup of G. Then µ is normal if and only if µ([x, y]) ≥ µ(x)∀x, y ∈ G. Proof. Suppose µ is a normal fuzzy subgroup of G. Let x, y ∈ G. Then µ(x−1 y −1 xy) µ(x−1 ) ∧ µ(y −1 xy) = µ(x) ∧ µ(x) = µ(x). Conversely, assume that µ satisﬁes the inequality. Then for all x, z ∈ G, we have µ(x−1 zx) = µ(zz −1 x−1 zx) ≥ µ(z) ∧ µ([z, x]) = µ(z) Thus µ(x−1 zx) µ(z)∀z, x ∈ G. Hence by Theorem 1.3.1 and Deﬁnition 1.3.2, µ is normal. We now give an example illustrating the above ideas. Example 2.1.5. Let G be the group of all symmetries of a square. Then G is a group of order 8 generated by a rotation of π/2 radians and a reﬂection along a diagonal of the square. Denote the elements of G by {e, π90 , π180 , π270 , h, v, d1 , d2 }, where e is the identity, π90 , π180 , π270 are

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rotations through 90, 180, 270 degrees, respectively, h and v are reﬂections about the horizontal and vertical axes, respectively, and d1 , d2 are reﬂections about the diagonals. It then follows that the conjugacy classes of G are {e}, {π180 }, {π90 , π270 }, {h, v}, {d1 , d2 }. Let H = {e, π180 } and K = {e, π90 , π180 , π270 }. Then H and K are normal subgroups of G and in fact, H is the center of G. Thus we have a chain of normal subgoups given by {e} ⊂ H ⊂ K ⊂ G. Now we construct a fuzzy subgroup of G whose level subgoups are the members of the above chain. Let ti ∈ [0, 1], i = 0, 1, 2, 3, be such that t0 > t1 > t2 > t3 . Deﬁne µ : G → [0, 1] as follows: ∀x ∈ G, µ(e) = t0 , µ(x) = t1 if x ∈ H\{e}, µ(x) = t2 if x ∈ K\H, µ(x) = t3 if x ∈ G\K = t3 . Then by Theorem 2.1.3, µ is a normal fuzzy subgroup of G. Furthermore, it is clear that µ is constant on the conjugacy classes of G. The following result follows from Theorem 1.3.12. However, we give another proof of it. Theorem 2.1.6. Let µ be a normal fuzzy subgroup of G. Then µ∗ G. Deﬁne the fuzzy subset µ# of G/µ∗ as follows: ∀xµ∗ ∈ G/µ∗ , µ# (xµ∗ ) = µ(x). # Then µ is a normal fuzzy subgroup of G/µ∗ . On the other hand, if N G # # and µ# 1 is a normal fuzzy subgroup of G/N such that µ1 (xN ) = µ1 (N ) only when x ∈ N, then there is a normal fuzzy subgroup µ of G such that, in the above notation, µ∗ = N and µ# = µ# 1 . Proof. Since µ is a normal fuzzy subgroup of G, it follows from Theorem 1.3.4 that µ∗ G. Further, if xµ∗ = yµ∗ for some x, y ∈ G, then y −1 x ∈ µ∗ and so µ(y −1 x) = µ(e). By Lemma 2.1.2, µ(x) = µ(y) and so µ# (xµ∗ ) = µ# (yµ∗ ). Therefore, µ# is well-deﬁned. It follows easily that µ# is a fuzzy subgroup of G/µ∗ We now show that µ# is normal. Let x, y ∈ G. Then µ# (xµ∗ yµ∗ ) = µ# (xyµ∗ ) = µ(xy) = µ(yx) (since µ is normal) = µ# (yµ∗ xµ∗ ). Hence µ# is normal. Conversely, given the normal fuzzy subgroup µ# 1 of G/N. Let µ be the fuzzy # subset of G deﬁned as follows: ∀x ∈ G, µ(x) = µ1 (xN ). It follows easily that µ is a fuzzy subgroup of G. Let x, y ∈ G. Then −1 N) µ(yxy −1 ) = µ# 1 (yxy # = µ1 (yN xN y −1 N ) # = µ# 1 (xN ) (since µ1 is normal) = µ(x). Hence µ is constant on the conjugacy classes of G and so by Theorem 2.1.3, µ is a normal fuzzy subgroup of G.

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Furthermore, if n ∈ N, we have that # µ(n) = µ# 1 (nN ) = µ1 (N ) = µ(e). # Thus N ⊆ µ∗ . Let x ∈ µ∗ . Then µ(x) = µ(e) and so µ# 1 (xN ) = µ1 (N ). Consequently, x ∈ N. Thus µ∗ ⊆ N. Therefore, N = µ∗ . It now follows that # µ# 1 =µ . Proposition 2.1.7. If µ is a normal fuzzy subgroup of G, then for all x ∈ G, xµ(xz) = xµ(zx) = µ(z) ∀z ∈ G. Proof. By Theorem 2.1.3, since µ is normal, µ is constant on the conjugacy classes of G. Hence by the deﬁnition of xµ and the fact that µ(xz) = µ(zx), we get xµ(zx) = xµ(xz) = µ(z). Similarly, xµ(zx) = µ(z). Proposition 2.1.7 is analogous to the result in group theory that if N G, then N x = xN ∀x ∈ G. If N is a normal subgroup of a group G, then the cosets of G with respect to N form a group (called the quotient group G/N ). We now recall Theorem 1.3.12. Let µ be a normal fuzzy subgroup of G. Let G/µ be the set of all the fuzzy cosets of µ. Then G/µ is a group under the composition xµ ◦ yµ = (xy)µ ∀x, y ∈ G. Deﬁne the fuzzy subset µ(∗) of G/µ by µ(∗) (xµ) = µ(x) ∀x ∈ G. Then µ(∗) is a normal fuzzy subgroup of G/µ. Deﬁnition 2.1.8. Let µ be a normal fuzzy subgroup of G. Then the fuzzy subgroup µ(∗) of G/µ deﬁned above is called the fuzzy quotient group determined by µ. Theorem 2.1.9. Let µ be a normal fuzzy subgroup of G. Deﬁne the function θ : G → G/µ as follows: ∀x ∈ G, θ(x) = xµ. Then θ is a homomorphism with kernel µ∗ = {x ∈ G | µ(x) = µ(e)}. Proof. Let x, y ∈ G. Then θ(xy) = xyµ = xµ ◦ yµ = θ(x) ◦ θ(y). Hence θ is a homomorphism. Let H = {h ∈ G | hµ = eµ}. Then h ∈ Ker(θ) ⇔ θ(h) = eµ ⇔ hµ = eµ ⇔ h ∈ H. Thus Ker(θ) = H. Now h ∈ H ⇔ hµ(x) = eµ(x)∀x ∈ G ⇔ µ(h−1 x) = µ(x)∀x ∈ G ⇔ µ(h−1 ) = µ(e) (by Lemma 2.1.2) ⇔ µ(h) = µ(e) ⇔ h ∈ µ∗ . Thus Ker(θ) = µ∗ . Corollary 2.1.10. Let µ be a normal fuzzy subgroup of G. Then G/H = G/µ∗ G/µ. The following result follows from Theorems 1.3.14, 1.3.15 and 1.4.8. Theorem 2.1.11. Let µ be a normal fuzzy subgroup of G. Then every (normal) fuzzy subgroup of G/µ corresponds in a natural way to a (normal) fuzzy subgroup of G.

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Let µ be a fuzzy subgroup of a ﬁnite group G. Then clearly G/µ is a ﬁnite set. Deﬁnition 2.1.12. Let µ be a fuzzy subgroup of a ﬁnite group G. Then the cardinality of G/µ is called the index of µ in G, written [G : µ]. It is clear from Corollary 2.1.10 that the index of µ in G divides the order of G. However, in Corollary 2.1.10, µ is normal. In Section 2.3, we prove that the index of any fuzzy subgroup of a ﬁnite group divides the order of the group. Also, Corollary 2.1.10 provides the details to Theorem 1.3.12(3).

2.2 Characteristic Fuzzy Subgroups and Abelian Fuzzy Subgroups We prove a number of results about fuzzy groups involving the concepts of fuzzy cosets and fuzzy normal subgroups. These results are analogs of standard results from group theory. Also, we introduce analogs of some grouptheoretic concepts such as characteristic subgroups. We prove that if µ is a fuzzy subgroup of a group G such that the fuzzy index of µ is the smallest prime dividing the order of G, then µ is a normal fuzzy subgroup. Deﬁnition 2.2.1. Let µ be a fuzzy subgroup of a group G and θ a function from G into itself. Deﬁne the fuzzy subset µθ of G by, ∀x ∈ G,

µθ (x) = µ(xθ ), where xθ = θ(x).

For a group G, a subgroup K is called a characteristic subgroup if K θ = K for every automorphism θ of G, where K θ denotes θ(K). We now deﬁne an analog of this notion. Deﬁnition 2.2.2. A fuzzy subgroup µ on a group K is called a fuzzy characteristic subgroup of G if µθ (x) = µ(x) for every automorphism θ of G and all x ∈ G. Theorem 2.2.3. Let µ be a fuzzy subgroup of a group G. Then the following assertions hold. (1) If θ is a homomorphism of G into itself, then µθ is a fuzzy subgroup of G. (2) If µ is a fuzzy characteristic subgroup of G, then µ is a normal.

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Proof. (1) Let x, y ∈ G. Then µθ (xy) = µ((xy)θ ) = µ(xθ y θ ) since θ is a homomorphism. Since µ is a fuzzy subgroup of G, µ(xθ y θ ) µ(xθ ) ∧ µ(y θ ) = µθ (x) ∧ µθ (y). Thus µθ (xy) µθ (x) ∧ µθ (y). Also, µθ (x−1 ) = µ((x−1 )θ ) = µ((xθ )−1 ) = µ(xθ ) = µθ (x). Hence µθ is a fuzzy subgroup of G. (2) Let x, y ∈ G. To prove that µ is normal, we have to show that µ(xy) = µ(yx). Let θ be the function of G into itself deﬁned by θ(z) = x−1 zx ∀z ∈ G. It is a standard result that θ is an automorphism of G (called the inner automorphism induced by x). Now since µ is a fuzzy characteristic subgroup of G, µθ = µ. Thus µ(xy) = µθ (xy) = µ((xy)θ ) = µ(x−1 (xy)x) = µ(yx). Hence µ is normal. Theorem 2.2.3 (2) is an analog of the result that a characteristic subgroup of a group is normal. We now obtain some other analogs. Deﬁnition 2.2.4. Let µ1 , µ2 be two fuzzy subgroups of a group G. We say that µ1 is conjugate to µ2 if there exists y ∈ G such that ∀x ∈ G, µ1 (x) = µ2 (y −1 xy). Clearly, if µ1 and µ2 are fuzzy subgroups of a group G such that µ1 is conjugate to µ2 , then µ1 and µ2 are conjugate fuzzy subgroups of G as deﬁned previously. It is easy to verify that the relation of conjugacy is an equivalence relation on the set of all fuzzy subgroups of a group. Consequently, the set of all fuzzy subgroups of a group is a union of pairwise disjoint classes of fuzzy subgroups each consisting of fuzzy subgroups which are equivalent to one another. We now obtain an expression giving the number of distinct conjugates of a fuzzy subgroup. First we give some preliminaries. If µ is a fuzzy subgroup of a group G and g ∈ G, recall that we denote by µg the fuzzy subset of G deﬁned by: µg (u) = µ(g −1 ug)∀u ∈ G. Then from Theorem 2.2.3 (1), it follows that µg is a fuzzy subgroup of G. Let g ∈ G and let θ be the automorphism of G deﬁned by θ(x) = g −1 xg ∀x ∈ G. Then µg (x) = µ(g −1 xg) = µ(θ(x)) = µθ (x) ∀x ∈ G. Hence µg = µθ .

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Recall that if µ is a fuzzy subgroup of a group G, the normalizer of µ is the set given by N (µ) = {g ∈ G | µ(gy) = µ(yg) ∀y ∈ G}. We now note that N (µ) = {g ∈ G | µg = µ} : It suﬃces to show that µ(gy) = µ(yg) ∀y ∈ G ⇔ µ(g −1 yg) = µ(y) ∀y ∈ G. We have that µ(g −1 yg) = µ(y) ∀y ∈ G ⇔ µ(g −1 (gy)g) = µ(gy) ∀y ∈ G ⇔ µ(yg) = µ(gy) ∀y ∈ G. Theorem 2.2.5 (2) below illustrates the motivation behind the term “normalizer”, and it also shows the analogy with the fact that a subgroup H of a group G is normal in G if and only if the normalizer of H in G is equal to G itself. The reader may wish to compare the proof of Theorem 2.2.5 with that of Theorems 1.3.7 and 1.3.8. Theorem 2.2.5. Let µ be a fuzzy subgroup of a group G. Then the following assertions hold. (1) N (µ) is a subgroup of G. (2) µ is normal if and only if N (µ) = G. (3) The number of distinct conjugates of µ is equal to the index of N (µ) in G, provided that G is a ﬁnite group. Proof. (1) Let g, h ∈ N (µ). Then µgh = (µg )h = µh = µ. Hence gh ∈ N (µ). Let g ∈ N (µ). We show that g −1 ∈ N (µ). For all y ∈ G, µ(gy) = µ(yg) and so µ((gy)−1 ) = µ((yg)−1 ). Thus ∀y ∈ G, µ(y −1 g −1 ) = µ(g −1 y −1 ) and so µ(yg −1 ) = µ(g −1 y), where the latter equality holds since y and y −1 are equally arbitrary. Thus g −1 ∈ N (µ). Hence N (µ) is a subgroup of G. (2) Let µ be normal and g ∈ G. Then for all u ∈ G, we have µg (u) = µ(g −1 ug) = µ((g −1 u)g) = µ(g(g −1 u)) (since µ is normal) = µ(u). Thus µg = µ and so g ∈ N (µ). Therefore, N (µ) = G. Conversely, suppose N (µ) = G. Let x, y ∈ G. To prove that µ is normal, we show that µ(xy) = µ(yx). Now µ(xy) = µ(xyxx−1 ) = µ(x(yx)x−1 ) −1 = µx (yx) = µ(yx), where the last equality follows since N (µ) = G and so x−1 ∈ N (µ). Hence −1 µx = µ. Thus µ is normal. (3) The proof of this result is based on the same technique used to prove the corresponding result for groups. Consider the decomposition of G,

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G = x1 N (µ) ∪ x2 N (µ) ∪ ... ∪ xk N (µ) as a union of pairwise disjoint cosets of N (µ), where k is the number of distinct cosets, that is, the index of N (µ) in G. Let x ∈ N (µ) and i ∈ N be such that 1 i k. Then for all g ∈ G, µxi x (g) = µ((xi x)−1 g(xi x)) = µ(x−1 (x−1 i gxi )x) = µx (x−1 i gxi ) (since x ∈ N (µ)) = µ(x−1 i gxi ) = µxi (g). Thus we have, µxi x = µxi ∀x ∈ N (µ), 1 i k. Hence any two elements of G which lie in the same coset xi N (µ) give rise to the same conjugate µxi of µ. Now we show that two distinct cosets give two distinct conjugates of µ. For this, suppose that µxi = µxj , where j = i and 1 i, j k. Thus µxi (g) = µxj (g) ∀g ∈ G, −1 µ(x−1 i gxi )= µ(xj gxj ) ∀g ∈ G. −1 Choose g = xj hxj . Then −1 −1 −1 µ(x−1 ∀h ∈ G i xj hxj xi ) = µ(xj xj hxj xj ) −1 −1 ⇒ µ((x−1 x ) h(x x )) = µ(h) ∀h ∈G i i j j −1

∀g ∈ G ⇒ µxj xi (h) = µ(h) −1 ⇒ xj xi ∈ N (µ). ⇒ xi N (µ) = xj N (µ). However, if i = j this is not possible since {x1 N (µ), x2 N (µ), ..., xk N (µ)} is a partition of G. Hence the number of distinct conjugates of µ is equal to the index of N (µ) in G. We now state a result which we use in a number of places. Its proof appears in the proof of Theorem 2.1.9. In this proof, the normality of µ is not used. Lemma 2.2.6. Let µ be a fuzzy subgroup of a ﬁnite group G. Let K = {x ∈ G|xµ = eµ}. Then K is a subgroup of G. In fact, K = µ∗ . It is a standard result in group theory that if G is a group and H, K are subgroups of G, H normal in G, then H ∩ K is normal in K. We now derive an analog for fuzzy subgroups. Proposition 2.2.7. Let µ be a fuzzy subgroup of G and ν be a normal fuzzy subgroup of G. Then (µ ∩ ν)|µ∗ is a normal fuzzy subgroup of the subgroup µ∗ . Proof. By Corollary 1.2.7, µ∗ is a subgroup of G. By Theorem 1.2.12, we have that µ ∩ ν is a fuzzy subgroup of G. Therefore, (µ ∩ ν)|µ∗ is a fuzzy subgroup

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of µ∗ . We now show that (µ ∩ ν)|µ∗ is a normal fuzzy subgroup of µ∗ . Let x, y ∈ µ∗ . Then xy and yx belong to µ∗ since µ∗ is a group. Hence µ(xy) = µ(yx) = µ(e) by the deﬁnition of µ∗ . Now (µ ∩ ν)(xy) = µ(xy) ∧ ν(xy) = µ(yx) ∧ ν(yx) = (µ ∩ ν)(yx). Hence (µ ∩ ν)|µ∗ is a normal fuzzy subgroup of µ∗ . If H is a subgroup of a group G, then H is normal in G if and only if Hx = xH ∀x ∈ G. As an analog, we prove the following result. Proposition 2.2.8. Let µ be a fuzzy subgroup of a group G. Then µ is normal if and only if ∀x ∈ G, µx = xµ. Proof. Suppose µ is normal. Then for all x, y ∈ G, µx(y) = µ(yx−1 ) = µ(x−1 y) = xµ(y). Thus µx = xµ ∀x ∈ G. Conversely, suppose that µx = xµ for all x in G. Now µ(xy) = (x−1 µ)(y) = (µx−1 )(y) = µ(yx) ∀x, y in G. Hence µ is normal. We restrict ourselves in the subsequent discussion to fuzzy left cosets only. Corresponding results for fuzzy right cosets could be obtained easily. Consequently, from now on we call a fuzzy left coset a “fuzzy coset” and denote it as xµ for all x ∈ G. It is a well-known result in group theory that subgroup of index 2 is a normal subgroup. We prove here an analog of a generalization of this result. Theorem 2.2.9. If µ is a fuzzy subgroup of a ﬁnite group G such that the fuzzy index of µ is p, the smallest prime dividing the order of G, then µ is a normal fuzzy subgroup of G. Proof. By Corollary 1.2.7, µ∗ is a subgroup of G. Further, by Lemma 2.2.6 and [[9], Theorem 4.10] µ∗ has index p in G, that is, µ∗ has p distinct (left) cosets, say, {xi µ∗ | 1 i p}. For all x ∈ G, deﬁne the function πx of G/µ∗ into itself by ∀g ∈ G, πx (gµ∗ ) = xgµ∗ . Then πx is a one-to-one function of G/µ∗ onto itself. Let P (G) = {πx | x ∈ G}. Now consider the permutation representation of G on the cosets of µ∗ given by the function π of G onto P (G), where π(x) = πx ∀x ∈ G. As is well known, π is an isomorphism of G into the symmetric group Sym(p), since the index of µ∗ in G is p. Furthermore, the kernel of the map π is the core of µ∗ , written Core(µ∗ ),that is, the intersection of all the conjugates g −1 µ∗ g, g ∈ G. By the fundamental theorem of homomorphism of groups and using Lagrange’s theorem, we have that the order of G/Core(µ∗ ) divides p!, which is the order of Sym(p). Since G/µ∗ ∼ = (G/Core(µ∗ ))/(µ∗ /Core(µ∗ )), |G/Core(µ∗ )| = |G/µ∗ ||µ∗ /Core(µ∗ )|. Now the order of G/µ∗ is p. Thus it follows that the order of µ∗ /Core(µ∗ ) divides (p − 1)!. Now since the order of µ∗ divides the order of G, we obtain that µ∗ = Core(µ∗ ), otherwise we get a contradiction of the fact that p is

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the smallest prime dividing the order of G. Since Core(µ∗ ) is always a normal subgroup of G, it follows that µ∗ is a normal subgroup of G. Now consider the quotient group G/µ∗ . Since the order of G/µ∗ is p, G/µ∗ is Abelian. Hence for all x, y ∈ G, we have (xµ∗ )(yµ∗ ) = (yµ∗ )(xµ∗ ) ⇒ xyµ∗ = yxµ∗ ⇒ xy = yxh for some h ∈ µ∗ . Hence µ(xy) = µ(yxh) µ(yx) ∧ µ(h) µ(yx) ∧ µ(e) = µ(yx). Thus µ(xy) µ(yx). Similarly, we obtain that µ(yx) µ(xy) Hence µ(xy) = µ(yx) ∀x, y ∈ G. Therefore, µ is normal. Corollary 2.2.10. If µ is a fuzzy subgroup of G such that the fuzzy index of µ is 2, then µ is normal. We now derive an analog of the following result from group theory which states that if θ is a homomorphism of a group G into itself whose kernel is N, then θ induces a homomorphism from G/N into itself. Theorem 2.2.11. Let µ be a normal fuzzy subgroup of G and let θ be a homomorphism of G into itself which leaves invariant the subgroup µ∗ . Then θ induces a homomorphism θ of G/µ into itself deﬁned by θ(xµ) = θ(x)µ ∀x ∈ G. Proof. First we show that θ is well deﬁned. To show this, suppose x, y ∈ G are such that xµ = yµ. Then it suﬃces to prove that θ(x)µ = θ(y)µ. Since xµ = yµ, we have that xµ(x) = yµ(x), xµ(y) = yµ(y) ⇒ µ(e) = µ(y −1 x), µ(x−1 y) = µ(e) ⇒ µ(y −1 x) = µ(x−1 y) ⇒ y −1 x, x−1 y ∈ µ∗ . Since by hypothesis θ(µ∗ ) = µ∗ , we get that θ(y −1 x) and θ(x−1 y) also belong to µ∗ . Thus we have µ(θ(y −1 x)) = µ(θ(x−1 y)) = µ(e). Now let g ∈ G. Then θ(x)µ(g) = µ(θ(x−1 )g) = µ(θ(x−1 )θ(y)θ(y −1 )g) µ(θ(x−1 )θ(y)) ∧ µ(θ(y −1 )g)

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= µ(θ(x−1 y)) ∧ θ(y)µ(g) = µ(e) ∧ θ(y)µ(g) = θ(y)µ(g). Thus θ(x)µ(g) θ(y)µ(g). Similarly, we have that θ(x)µ(g) ≤ θ(y)µ(g). Since g ∈ G is arbitrary, we have θ(x)µ = θ(y)µ. Therefore, θ is well deﬁned. Next we show that θ is a homomorphism. Let x, y ∈ G. Since θ is a homomorphism, θ(xy) = θ(x)θ(y). Thus θ(xy)µ = θ(x)θ(y)µ. Hence θ(xyµ) = θ(x)µ ◦ θ(y)µ. Thus θ(xµ ◦ yµ) = θ(xµ) ◦ θ(yµ.) Therefore, θ is a homomorphism. In Theorem 2.2.11, we have assumed µ to be normal instead of assuming only that µ is a fuzzy subgroup. This has been done to ensure that the law of composition of fuzzy cosets is well deﬁned. This fact is used in the proof of Theorem 2.2.11 to show that θ is a homomorphism. However, it is clear from the proof that to show θ is well deﬁned it is not necessary to assume µ to be normal. Corollary 2.2.12. With the same hypothesis as in Theorem 2.2.11, the homomorphism θ is an automorphism if θ is an automorphism and G is ﬁnite. Proof. Since G has ﬁnite order, it is easy to see that θ has ﬁnite order. Suppose that θ has order k. Then θk = ι, where ι denotes the identity map. Now since by Theorem 2.2.11, we have that θ is a homomorphism, it remains to prove that θ is one-to-one. For this purpose, let x, y ∈ G be such that θ(xµ) = θ(yµ). Then θ(x)µ = θ(y)µ. Thus θ(θ(x)µ) = θ(θ(y)µ). This implies by deﬁnition of θ that θ2 (x)µ = θ2 (y)µ. Iterating, we obtain that ι(x)µ = θk (x)µ = θk (y)µ = ι(y)µ and so xµ = yµ. Hence θ is one-to-one. Corollary 2.2.13. With the same hypothesis as in Theorem 2.2.11, the function θ is an automorphism of G if θ is an automorphism and µ∗ = {e}. Proof. Let x, y ∈ G be such that θ(x) = θ(y). Then it follows that θ(x)µ = θ(y)µ That is, θ(xµ) = θ(yµ). Hence, xµ = yµ since θ is one-to-one by the hypothesis. Thus xµ(y) = yµ(y), which implies that µ(x−1 y) = µ(e). Therefore, x−1 y ∈ µ∗ and so x−1 y = e since µ∗ = {e} by the hypothesis. Thus x = y. Hence θ is one-to-one.

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We now consider a variation of Corollary 2.2.13. The motivation of the following result arises from the standard theorem in group theory that says that if θ is an automorphism of G and N is a normal subgroup of G such that N θ ⊆ N, then θ induces an automorphism of the quotient group G/N. We prove the following result. Theorem 2.2.14. Let µ be a normal fuzzy subgroup of G and θ be an automorphism of G such that µθ = µ. Then θ induces an automorphism θ of G/µ deﬁned by θ(xµ) = θ(x)µ ∀x ∈ G. Proof. Let x, y ∈ G. Then xµ = yµ ⇔ xµθ = yµθ ⇔ xµθ (g) = yµθ (g)∀g ∈ G ⇔ µθ (x−1 g) = µθ (y −1 g)∀g ∈ G ⇔ µ(θ(x−1 g)) = µ(θ(y −1 g))∀g ∈ G ⇔ µ(θ(x−1 )θ(g)) = µ(θ(y −1 )θ(g))∀g ∈ G ⇔ θ(x)µ(θ(g)) = θ(y)µ(θ(g))∀g ∈ G ⇔ θ(x)µ = θ(y)µ (since θ is an automorphism) ⇔ θ(xµ) = θ(yµ). Thus θ is well deﬁned and one-to-one. Clearly, θ maps G/µ onto itself. The proof of the fact that θ is a homomorphism is analogous to the corresponding part of the proof of Theorem 2.2.11 and we omit the details. The next result gives a relationship between fuzzy cosets of a fuzzy subgroup and the cosets of a subgroup of the given group. Recall Theorem 1.3.10. Let µ be a fuzzy subgroup of a ﬁnite group G. Then for x, y ∈ G, we have that xµ∗ = yµ∗ ⇔ xµ = yµ. Thus [G : µ] = [G : µ∗ ]. This shows that there is a natural one-to-one correspondence between the (left) cosets of µ∗ in G and the fuzzy cosets of µ in G. Thus we see that the subgroup µ∗ plays a key role in the analysis of fuzzy cosets.

2.3 Fuzzy Caley’s Theorem and Fuzzy Lagrange’s Theorem In this section, we present the fuzzy analogs of Cayley’s Theorem and Lagrange’s Theorem. We develop here the notions of the “order” of a fuzzy group and fuzzy Abelian group. We also introduce the concept of a fuzzy solvable group. We obtain some analogs of group theoretical results related to these ideas introduced here. Recall that if µ is a fuzzy subgroup of G and x ∈ G, then µ(xy) = µ(y) ∀y ∈ G ⇔ µ(x) = µ(e). Recall also that if µ is a normal fuzzy subgroup of G, then G/µ is a group under the operation : xµ ◦ yµ = xyµ ∀x, y ∈ G.

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Furthermore, if µ ¯ is the fuzzy subset of G/µ deﬁned by µ ¯(xµ) = µ(x) ∀x ∈ G, then µ ¯ is a fuzzy subgroup of G/µ. Let µ be a normal fuzzy subgroup of ˆ of a group G. Then µ∗ is a normal subgroup of G. Deﬁne the fuzzy subset µ ˆ(xH) = µ(x)∀x ∈ G. Then µ ˆ is a normal fuzzy subgroup of G/µ∗ as follows: µ G/µ∗ . Let µ be a fuzzy subgroup of G. Then µ∗ = {x ∈ G | xµ = eµ}. We now present a fuzzy Caley’s Theorem. Much of its proof has been seen in Theorems 1.3.12 and 2.2.11. We note that we have the usual Caley’s theorem if we let µ = 1{e} in the next result. Theorem 2.3.1. Let µ be a normal fuzzy subgroup of a group G. Then there exists a homomorphism π of G onto G/µ that induces a natural permutation representation of G/µ. Furthermore, the kernel of π is µ∗ . Proof. For all x ∈ G, deﬁne a function πx : G/µ → G/µ by ∀y ∈ G, πx (yµ) = xyµ. We now show that ∀x ∈ G, πx is a permutation of G/µ. Let x ∈ G. We have that πx (yµ) = πx (zµ) ⇔ xyµ = xzµ ⇔ xyµ(g) = xzµ(g) ∀g ∈ G ⇔ µ(y −1 x−1 g) = µ(z −1 x−1 g) ∀g ∈ G ⇔ µ(y −1 u) = µ(z −1 u) ∀u ∈ G (since x−1 g is also an arbitrary element of G) ⇔ yµ(u) = zµ(u)∀u ∈ G ⇔ yµ = zµ. Thus πx is a one-to-one function of G/µ into itself ∀x ∈ G. Thus ∀x ∈ g, πx is a permutation of G/µ since it follows easily that πx maps G/µ onto itself. Now let Sym(G/µ) denote the symmetric group on G/µ (that is, the group of all permutations of G/µ. Let π : G →Sym(G/µ) be the function deﬁned by π (x) = πx ∀x ∈ G. We show that π is a homomorphism. Let x, y ∈ G. Then for all g ∈ G, we have that πxy (gµ) = (xy)gµ = x(yg)µ = πx (ygµ) = πx (πy (gµ)) = πx ◦ πy (gµ) Thus πxy = πx ◦ πy . Hence π(xy) = π(x)π(y). Thus π is a homomorphism. We now prove that the kernel of π is µ∗ . x ∈ Ker(π) ⇔ π(x) = πe ⇔ πx = πe ⇔ πx (gµ) = πe (gµ)∀g ∈ G ⇔ xgµ = gµ∀g ∈ G ⇔ xµgµ = gµ∀g ∈ G ⇔ xµ = eµ ⇔ xµ(g) = eµ(g)∀g ∈ G ⇔ µ(x−1 g) = µ(g)∀g ∈ G ⇔ µ(x−1 ) = µ(e) ⇔ µ(x) = µ(e) ⇔ x ∈ µ∗ . Therefore, Ker(π) = µ∗ . In the next result, we present another type of fuzzy Caley’s Theorem. Let G be a group and ∀x ∈ G deﬁne the function fx of G into itself by ∀y ∈ G, fx (y) = xy. Then it is known that fx is a permutation of G. Let P (G) = {fx | x ∈ G}. Then P (G) is a group under composition of functions and G ∼ = P (G) under the isomorphism f such that f (x) = fx ∀x ∈ G. This result is Caley’s Theorem. Theorem 2.3.2. Let µ be a fuzzy subgroup of a group G. Let S(µ) denote the set of fuzzy singletons {xt | xt ⊆ µ, x ∈ G}. Let f be the isomorphism of G onto P (G) such that f (x) = fx ∀x ∈ G. Then S(µ) is a completely regular semigroup with identity under max-min composition and f induces an isomorphism of S(µ) onto the semigroup S(f (µ)) = {(fx )t | 0 ≤ t ≤ µ(x)}.

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Proof. Clearly, S(µ) is a semigroup with identity eµ(e) . In fact, S(µ) is completely regular since it is the disjoint union of groups {xt | x ∈ µt } for 0 ≤ t ≤ µ(e). Let x ∈ G. Since f (x) = fx , f (xt )(fy ) = ∨{xt (z) | f (z) = fy } = t if f (x) = fy , i.e., fx = fy and 0 otherwise. Thus f (xt ) = (fx )t . Now f (xy) = f (x)f (y) for all x, y ∈ G. Hence f (xr ys ) = f ((xy)r∧s ) = f (xy)r∧s = (fxy )r∧s = (fx ◦ fy )r∧s = (fx )r (fy )s . Since f is one-to-one, xr = ys ⇔ x = y and r = s ⇔ f (x) = f (y) and r = s ⇔ fx = fy and r = s ⇔ (fx )r = (fy )s ⇔ f (xr ) = f (ys ). We now develop the concept of fuzzy order. We note that Example 4.1.12 shows that the converse of the next result does not hold in general. Proposition 2.3.3. Let µ be a fuzzy subgroup of a group G. If µ([x, y]) = µ(e)∀x, y ∈ G, then µ is normal in G. Proof. Recall [x, y] = x−1 y −1 xy. We have shown earlier that µ is normal if and only if µ is constant on the conjugacy classes of G. Thus we have that µ is normal ⇔ µ(x−1 y −1 x) = µ(y −1 ) ∀x, y ∈ G ⇔ µ(x−1 y −1 xyy −1 ) = µ(y −1 ) ∀x, y ∈ G ⇔ µ([x, y] y −1 ) = µ(y −1 ) ∀x, y ∈ G ⇐ µ([x, y]) = µ(e) ∀x, y ∈ G by Lemma 2.1.2.

(Note that [x, y] does not remain ﬁxed as y varies.)

Corollary 2.3.4. Let µ be a fuzzy subgroup of a group G. If µ([x, y]) = µ(e)∀x, y ∈ G, then µ∗ is a normal subgroup of G. Furthermore, the quotient group G/µ∗ is Abelian. Proof. The fact that µ∗ is a normal subgroup of G follows from Proposition 2.3.3 and Theorem 1.3.4. Let G denote the commutator subgroup of G, i.e., the subgroup generated by all the [x, y] for x, y ∈ G. Since µ([x, y]) = µ(e) ∀x, y ∈ G, G ⊆ µ∗ . Consequently, G/µ∗ is Abelian. Now we describe the motivation behind the two concepts of the order of a fuzzy subgroup and that of a fuzzy Abelian subgroup of a group G. Let K be a subgroup of a group G. Consider the set H = {x ∈ G | 1K (x) = 1K (e)}. Recall that for any subgroup M of G, the index of M in G, denoted by [G : M ] , is equal to the number of distinct (left) cosets of M in G, which is also equal to o(G)/o(M ), where o(G) denotes the order of G. Now in the above situation it is clear that H = K, and the index of K in G is indeed equal to the index of H in G. Furthermore, if [G : H] = r, then we have that

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o(H) = o(K) = o(G) r . Also, we note that H is Abelian if and only if K is Abelian. These considerations motivate us to consider the following analogs in the case of a fuzzy subgroup µ of G. Deﬁnition 2.3.5. Let µ be a fuzzy subgroup of a ﬁnite group G. Then the order of µ, written o(µ), is deﬁned to be o(µ) = o(G) r , where r is the index of µ. Deﬁnition 2.3.6. Let µ be a fuzzy subgroup of a ﬁnite group G. Then µ is called fuzzy Abelian if µ∗ is an Abelian subgroup of G. Now µ∗ is always a subgroup of G, but it is not necessarily Abelian. It follows that o(G) . o(µ) = [G : µ∗ ] Hence o(µ) = o(µ∗ ). If H, K are subgroups of a ﬁnite group G such that o(H) = o(K) and H ⊆ K, then trivially H = K. However, if µ, ν are fuzzy subgroups of a ﬁnite group G such that o(µ) = o(ν) and µ ⊆ ν, then it is always not necessarily the case that µ = ν, as the following example shows. Example 2.3.7. Let G = {e, a, b, c} be the Klein 4-group, where e is the identity of G. Let t0 , t1 , t2 ∈ [0, 1] be such that t 0 > t1 > t2 . Deﬁne ν : G → [0, 1] as follows: ν(e) = t0 , ν(a) = t1 , ν(b) = t2 , ν(ab) = t2 . It follows easily that ν is a fuzzy subgroup of G. Let s0 , s1 , s2 ∈ [0, 1] be such that s0 > s1 > s2 , where t0 < s0 , t1 < s1 , t2 < s2 . Deﬁne µ : G → [0, 1] as follows: µ(e) = s0 , µ(a) = s1 , µ(b) = s2 , µ(ab) = s2 . It again follows easily that µ is a fuzzy subgroup of G. Furthermore, it follows that ν ⊆ µ. From the deﬁnitions of ν and µ, it is clear that ν∗ = µ∗ . Thus by using (13) we get that o(ν) = o(ν∗ ) = o(µ). Hence we see that ν and µ have the same orders, ν ⊆ µ, and yet clearly ν = µ.

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The above example illustrates the fact that not all properties of a fuzzy subgroup could be expected to be analogs of results from group theory. The subgroup ν deﬁned in Example 2.3.7 is also an example of a fuzzy Abelian subgroup, and the same is the case with the fuzzy subgroup µ deﬁned therein. Since a fuzzy subgroup ν of a group G is called fuzzy Abelian if ν∗ is Abelian, examples of fuzzy subgroups which are not fuzzy Abelian are easily constructed. For example, let G be any group containing a subgroup H that is not Abelian. Deﬁne the fuzzy subset µ of G by µ(x) = t0 ∀x ∈ H and µ(x) = t1 ∀x ∈ G\H, where 0 ≤ t1 < t0 ≤ 1. Then µ is a fuzzy subgroup of G and µ∗ = H. Thus µ is not fuzzy Abelian. We now prove an analog of a result from group theory that if U, V are two subgroups of a group G such that U ⊆ V and V is Abelian, then U is Abelian. We have the following result. Proposition 2.3.8. Let µ, ν be fuzzy subgroups of a group G such that ν ⊆ µ, ν(e) = µ(e), and µ is fuzzy Abelian. Then ν is fuzzy Abelian. Proof. By hypothesis µ∗ is Abelian. Let y ∈ ν∗ . Then we have that ν(y) = ν(e) = µ(e). By hypothesis, ν(y) ≤ µ(y). Since µ(y) ≤ µ(e), µ(y) = µ(e) and so y ∈ µ∗ . Thus ν∗ ⊆ µ∗ and consequently ν∗ is Abelian. Hence, ν is fuzzy Abelian. Proposition 2.3.9. A fuzzy subgroup of order p2 is fuzzy Abelian if p is a prime. Proof. Let µ be a fuzzy subgroup of group G such that o(µ) = p2 . By (13) we have o(µ) = o(µ∗ ). Now it is a standard result in group theory that a group of order p2 is Abelian. Thus µ∗ is Abelian and consequently µ is fuzzy Abelian. Let µ and ν be fuzzy subgroups of G such that ν ⊆ µ. Since o(µ) = o(µ∗ ), o(ν) = o(ν∗ ), [G : µ] = [G : µ∗ ], and [G : ν] = [G : ν∗ ], we are motivated to deﬁne the index of ν in µ, written [µ : ν], to be [µ∗ : ν∗ ]. The following theorem is one of several results one could call a Fuzzy Lagrange’s Theorem. Theorem 2.3.10. Let µ and ν be fuzzy subgroups of a ﬁnite group G such that ν ⊆ µ. Then the order of ν divides the order of µ. In fact, o(µ) = o(ν)[µ : ν]. Proof. As in the proof of Proposition 2.3.8, we have that ν∗ ⊆ µ∗ . Thus by Lagrange’s theorem, we have that o(µ∗ ) = o(ν∗ )[µ∗ : ν∗ ]. However, o(ν) = o(ν∗ ), o(µ) = o(µ∗ ), and [µ : ν] = [µ∗ : ν∗ ]. Hence the desired result holds. We now develop a fuzzy analog of solvability. First we need a preliminary result.

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Lemma 2.3.11. Let µ, ν be normal fuzzy subgroups of G such that ν ⊆ µ and ν(e) = µ(e). Then there exists a normal fuzzy subgroup η of the quotient group G/ν∗ such that η(xν∗ ) = µ(x) ∀x ∈ G. Proof. By Theorem 1.3.4, we have that ν∗ and µ∗ are normal subgroups of G. Since ν ⊆ µ, by arguing as in the proof of Proposition 2.3.8, we have that ν∗ ⊆ µ∗ . Deﬁne the fuzzy subset η of G/ν∗ by η(xν∗ ) = µ(x) ∀x ∈ G. We claim that η is well deﬁned. For suppose that xν∗ = yν∗ . Then y −1 x ∈ ν∗ and so y −1 x ∈ µ∗ since ν∗ ⊆ µ∗ . Consequently, we have that xµ∗ = yµ∗ . Now by Theorem 2.1.6, µ induces a unique function µ# from G/µ∗ into [0, 1] such that µ# (xµ∗ ) = µ(x) ∀x ∈ G. Therefore, we have that ∀x, y ∈ G, η(xν∗ ) = µ(x) = µ# (xµ∗ ) = µ# (yµ∗ ) = µ(y) = η(yν∗ ) since xµ∗ = yµ∗ . Thus η is well deﬁned. It follows easily that η is a fuzzy subgroup and that η is normal. Let µ, ν be normal fuzzy subgroups of G such that ν ⊆ µ. Then ν∗ is a normal subgroup of G. Let f be the natural homomorphism of G onto G/ν∗ . Then ∀x ∈ G, f (µ)(xν∗ ) = ∨{µ(y)|f (y) = xν∗ } = ∨{µ(y)|y ∈ ν∗ } = µ(x) = (µ/ν)(x). Hence f (µ) = µ/ν. Let x ∈ G. Then xν∗ ∈ (µ/ν)∗ ⇐⇒ (µ/ν)(xν∗ ) = (µ/ν)(eν∗ ) ⇐⇒ µ(x) = µ(e) ⇐⇒ x ∈ µ∗ . Thus (µ/ν)∗ = µ∗ /ν∗ . Deﬁnition 2.3.12. With the hypothesis of Lemma 2.3.11, the fuzzy subgroup η deﬁned above is called the fuzzy quotient group µ/ν. Now consider a chain of normal fuzzy subgroups of G given by ν1 ⊇ ν2 ⊇ ... ⊇ νk , where νi (e) = ν1 (e), i = 1, ..., k. Let Hi = (νi )∗ , i = 1, ..., k. Then we have that Hi is a normal subgroup of G since νi is normal. By Lemma 2.3.11, it follows that the chain ν1 ⊇ ν2 ⊇ ... ⊇ νk yields another chain of normal fuzzy subgroups: η1 ⊇ η2 ⊇ ... ⊇ ηk , where ηi = νi /νi+1 and Hi /Hi+1 = {xHi+1 | x ∈ G, ηi (x) = ηi (e)}. The above discussion motivates the following deﬁnition. Deﬁnition 2.3.13. A fuzzy subgroup ν of a group G is called fuzzy solvable if there exists a chain of normal fuzzy subgroups ν = ν1 ⊇ ν2 ⊇ ... ⊇ νk with νk (x) = νk (e) only when x = e, and νi (e) = ν1 (e), 1 ≤ i ≤ k, such that there is a corresponding chain of normal fuzzy subgroups η1 ⊇ η2 ⊇ ... ⊇ ηk , where ηi = νi /νi+1 and ηi is fuzzy Abelian, 1 ≤ i ≤ k, νk+1 = eν1 (e) . The chain of normal fuzzy subgroups given by ν = ν1 ⊇ ν2 ⊇ ... ⊇ νk yields a chain of normal subgroups H1 ⊇ H2 ⊇ ... ⊇ Hk of the group G, where Hi is deﬁned by Hi = (νi )∗ , i = 1, ..., k. Clearly Hk = {e}.

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If in the chain of fuzzy subgroups given by ν = ν1 ⊇ ν2 ⊇ ... ⊇ νk , we have that ν1 (x) = ν1 (e) ∀x ∈ G, then H1 = G. In this case, we have a chain of subgroups H1 = G ⊇ H2 ⊇ ... ⊇ Hk = {e} , where Hi /Hi+1 is Abelian. Consequently, G is solvable in this case. In the general situation when H1 is not necessarily equal to G, it is clear that H1 is always a solvable group. For a solvable group, it is a standard result that any subgroup and any quotient group are both solvable. For a fuzzy solvable fuzzy subgroup, we present one analog since we consider these issues in greater detail in Chapter 3. Theorem 2.3.14. Let G be a group. Let µ, ν be normal fuzzy subgroups of G such that ν ⊆ µ and ν(e) = µ(e). If µ is fuzzy solvable, then ν is fuzzy solvable. Proof. Since µ is solvable, there is a chain of normal fuzzy subgroups µ = µ1 ⊇ µ2 ⊇ ... ⊇ µk satisfying the conditions in Deﬁnition 2.3.13. Consider the chain of fuzzy subgroups ν = ν1 ⊇ µ2 ∩ ν ⊇ µ3 ∩ ν.... ⊇ µk ∩ ν. Clearly each µn ∩ ν is normal in G. Now x, y ∈ (µi ∩ ν)∗ ⇔ x, y ∈ (µi )∗ ∩ ν∗ . This implies that x, y ∈ (µi )∗ and x, y ∈ ν∗ which further implies that x−1 y −1 xy ∈ (µi+1 )∗ and x−1 y −1 xy ∈ ν∗ since (µi )∗ /(µi+1 )∗ is Abelian. Thus xy −1 xy ∈ (µi+1 )∗ ∩ ν∗ = (µi+1 ∩ ν)∗ and so (µi ∩ ν)∗ /(µi+1 ∩ ν)∗ is Abelian. Thus (µi ∩ ν)/(µi+1 ∩ ν) is fuzzy Abelian and so ν is fuzzy solvable. We next present another type of fuzzy Lagrange’s theorem using the notion of fuzzy order as developed in Section 1.6 and references [8, 9]. It follows that if {x ∈ G | µ(x) = µ(e)} = {e}, then F Oµ (x) = o(x) for all x ∈ G. For suppose µ(xn ) = µ(e) for some x ∈ G and some positive integer n. Then xn = e by assumption. Deﬁnition 2.3.15. Let G be a group and µ a fuzzy subgroup of G. If there exists a positive integer n such that for all x ∈ G, µ(xn ) = µ(e), then the smallest such positive integer is called the fuzzy order of µ, written O(µ). If no such positive integer exists, then µ is said to be of inﬁnite fuzzy order. It follows in Deﬁnition 2.3.15, O(µ) is the least common multiple of the F Oµ (x) for x ∈ G if this least common multiple exists. We also note that O(µ) = 1 if and only if µ = µ(e)G . Example 2.3.16. Consider the group Z under usual addition. Deﬁne the fuzzy subset µ of Z as follows: µ(x) = t0 if x ∈ 6 and µ(x) = t1 otherwise, where t0 > t1 . Then µ is a fuzzy subgroup of Z. If x ∈ 2 + 6 or x ∈ 4 + 6, then F Oµ (x) = 3. If x ∈ 3 + 6, then F Oµ (x) = 2. If x ∈ 6, then F Oµ (x) = 1. Also, O(µ) = 6.

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The following theorem provides a relation between fuzzy orders of fuzzy subgroups and orders of groups. It gives us another type of fuzzy Lagrange’s Theorem. Theorem 2.3.17. Let G be a ﬁnite group and µ a fuzzy subgroup of G. Then O(µ)|o(G). Proof. For all x ∈ G, F Oµ (x)|o(G) by Proposition 1.6.5. Thus o(G) is a common multiple of F Oµ (x) for all x ∈ G. Since O(µ) is the least common multiple of the F Oµ (x), it follows that O(µ)|o(G). Example 2.3.18. Let G = {e, a, b, c} be the Klein-4 group. Deﬁne the fuzzy subset µ of G as follows: µ(e) = 1, µ(a) = µ(b) = µ(c) = 1/2. Then µ is a fuzzy subgroup of G. Now O(µ) = 2 and [G : µ] = [G : µ∗ ] = 4. Hence O(µ)|o(G), but o(G) = O(µ)[G : µ]. Proposition 2.3.19. Let G be a ﬁnite group and µ a fuzzy subgroup of G. If O(µ) = pn m, where p is prime and m and n are relatively prime positive integers, then there is an x ∈ G such that F Oµ (x) = pk for each nonnegative integer k ≤ n. Proof. . Since O(µ) is the least common multiple of the F Oµ (x) for x ∈ G, there is an x ∈ G such that F Oµ (x) = pn . Hence the desired result follows easily. Corollary 2.3.20. Let G be a ﬁnite Abelian group and µ a fuzzy subgroup of G. Then the following assertions hold. (1) If O(µ) = mn for some positive integers m and n, then there is an x ∈ G such that F Oµ (x) = m. (2) O(µ) = ∨{F Oµ (x) | x ∈ G}. Proof. The desired result follows from Proposition 2.3.19 and Theorem 1.6.13(2). Lemma 2.3.21. Let G be a ﬁnite group and µ and ν fuzzy subgroups of G. If ν ⊆ µ and µ(e) = ν(e), then F Oµ (x)|F Oν (x) for all x ∈ G such that F Oν (x) is ﬁnite. Proof. Let F Oν (x) = n. Then µ(e) = ν(e) = ν(xn ) ≤ µ(xn ). Since µ(e) ≥ µ(xn ), µ(xn ) = µ(e). Thus F Oµ (x)|n by Proposition 1.6.5. We now obtain another fuzzy analogue of Lagrange’s theorem. The converse of the analogue does not hold in general although it holds for special cases. The converse of this analogue will be discussed in Section 7.2. Theorem 2.3.22. Let G be a ﬁnite group and µ and ν fuzzy subgroups of G. If ν ⊆ µ and µ(e) = ν(e), then O(µ)|O(ν).

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2 Fuzzy Caley’s Theorem and Fuzzy Lagrange’s Theorem

Proof. We may assume that O(ν) is ﬁnite. Then F Oν (x) is ﬁnite for all x ∈ G and the number of elements of H = {F Oν (x) | x ∈ G} is ﬁnite since G is ﬁnite. Thus F Oµ (x) is ﬁnite for all x ∈ G by Lemma 2.3.21 and the number of elements of K = {F Oµ (x) | x ∈ G} is ﬁnite. Thus O(ν) and O(µ) are the least common multiples of elements of H and K, respectively. Hence O(µ)|O(ν) by Lemma 2.3.21.

References 1. P. Bhattacharya, Fuzzy subgroups: Some characterization, J. Math. Anal. Appl., 128(1987) 241-252. 2. P. Bhattacharya, Fuzzy subgroups: Some characterizations II, Inform. Sci., 38(1986) 293-297. 3. P. Bhattacharya and N. P. Mukherjee, Fuzzy groups and fuzzy relations, Inform. Sci. 36 (1985) 267-282. 4. P. Bhattacharya and N. P. Mukherjee, Fuzzy groups: some group theoretic analogs. II, Inform. Sci. (1987) 77-91. 41 5. L. Biacino and G. Gerla, Closure systems and L-subalgebras, Inform. Sci. 33 (1984)181-195. 6. P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264-269. 7. W.B.V. Kandasamy and D. Meiyappan, Fuzzy symmetric subgroups and conjugate fuzzy subgroups of a group, Journal of Fuzzy Math. 6 (1998) 905-913. 8. J. G. Kim, Fuzzy orders relative to fuzzy subgroups, Inform. Sci 80 (1994) 341-348. 58 9. J. G. Kim, Orders of fuzzy subgroups and fuzzy p-subgroups, Fuzzy Sets and Systems 61 (1994) 225-230. 49, 58 10. N. P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225-239. 41 11. N. P. Mukherjee and P. Bhattacharya, Fuzzy groups: Some group theoretic analogs, Inform. Sci. 39 (1986) 247-268. 41 12. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. 13. L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. 14. Y. Zhang, Some properties on fuzzy subgroups, Fuzzy Sets and Systems 119 (2001) 427-438. 15. Y. Zhang and K. Zou, Normal fuzzy subgroups and conjugate fuzzy subgroups, Journal of Fuzzy Math. 1 (1993) 571-585.

3 Nilpotent, Commutator, and Solvable Fuzzy Subgroups

As the title suggests, this chapter is concerned with nilpotent, commutator, and solvable fuzzy subgroups. We give two approaches for the development of these notions, namely one via an ascending chain of fuzzy subgroups and one via descending chain of fuzzy subgroups. The results of this chapter are mainly from [5, 8, 18].

3.1 Commutative Fuzzy Subsets and Nilpotent Fuzzy Subgroups In this section, we introduce the notion of a commutative fuzzy subset and the notion of a nilpotent fuzzy subgroup. We show that the nilpotence of a group can be completely characterized by the nilpotence of its fuzzy subgroups. For the relevant classical group-theoretic results and deﬁnitions, the reader is referred to [6, 17]. Let µ be a fuzzy subset of a groupoid G. The normalizer N (µ) of µ in G is the set {x ∈ G | µ(xy) = µ(yx) for all y ∈ G} and µ is called normal in G if N (µ) = G. If G is a group, then µ(e) is called the tip of µ. Deﬁnition 3.1.1. Let µ be a fuzzy subset of a semigroup G. Let Z(µ) = {x ∈ G | µ(xy) = µ(yx) and µ(xyz) = µ(yxz) for all y, z ∈ G}. Then µ is called commutative in G if Z(µ) = G. We adopt the terminology from [8] and call Z(µ) in Deﬁnition 3.1.1 the centralizer of µ in G . If G has a right identity, then the equality µ(xy) = µ(yx) in Deﬁnition 3.1.1 is redundant. If G is a semigroup, we let Z(G) denote the center of G. It is clear that Z(G) ⊆ Z(µ) ⊆ N (µ) and that Z(G) = Z(µ) = N (µ) is possible. In the following example, recall that the notation a, b | a3 = e = b2 , ba = a2 b denotes the group generated by a and b, where a and b satisfy the properties a3 = e = b2 and ba = a2 b. John Mordeson, Kiran R. Bhutani, Azriel Rosenfeld: Fuzzy Group Theory, StudFuzz 182, 61–89 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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Example 3.1.2. Let G = D3 ⊗ D3 , where D3 is the dihedral group a, b | a3 = e = b2 , ba = a2 b . Let µ be the fuzzy subset of G deﬁned as follows: ∀x ∈ G,   t0 if x ∈ {e} ⊗ a , µ(x) = t1 if x ∈ (a ⊗ {e})\({e} ⊗ a),  t2 otherwise, where t0 > t1 > t2 . Since a is a normal subgroup of D3 , xy ∈ a if and only if yx ∈ a for all x, y ∈ D3 . This allows us to show that N (µ) = G. Let (x, y), (u, v) ∈ G. Then (xu, yv) ∈ a ⊗ {e} ⇔ (ux, vy) ∈ a ⊗ {e} and (xu, yv) ∈ {e} × {a} ⇔ (ux, vy) ∈ {e} ⊗ a. Hence (xu, yv) ∈ G\((a ⊗ {e}) ∪ ({e} ⊗ a)) ⇔ (ux, vy) ∈ G\((a ⊗ {e}) ∪ {e} ⊗ a)). Thus it follows that N (µ) = G. Now µ((e, a)(a, a2 b)(a, b)) = µ(a2 , e) = t1 and µ((a, a2 b)(e, a)(a, b)) = µ(a2 , a) = t2 . Hence Z(µ) = N (µ). Lemma 3.1.3. Let µ be a fuzzy subset of a semigroup G. Then x ∈ Z(µ) if and only if µ(xy1 , ..., yn ) = µ(y1 xy2 ...yn ) = ... = µ(y1 y2 ...yn x) for all y1 , y2 , ..., yn ∈ G. Proof. We prove the result by induction on n. Suppose x ∈ Z(µ). Then µ(xy1 y2 ) = µ(y1 xy2 ) for all y1 , y2 ∈ G. Assume µ(xy1 , ..., yn ) = µ(y1 xy2 ...yn ) = ... = µ(y1 y2 ...yn x) for all y1 , y2 , ..., yn ∈ G. Then µ(xy1 ...(yn yn+1 )) = µ(y1 xy2 ...(yn yn+1 )) = ... = µ(y1 y2 ...x(yn yn+1 )) = µ(y1 y2 ...(yn yn+1 )x) and µ(x(y1 y2 )...yn yn+1 ) = µ((y1 y2 )x...yn yn+1 ) = ... = µ((y1 y2 )...yn xyn+1 ) = µ((y1 y2 )...yn yn+1 x) for all y1 , ..., yn , yn+1 ∈ G. This yields the desired result. Theorem 3.1.4. Let µ be a fuzzy subset of a semigroup G. Then µ is commutative in G if and only if ∀x1 , ..., xn ∈ G, n ∈ N, µ(x1 ...xn ) = µ(xπ(1) ...xπ(n) ) for all permutations π of {1, ..., n}. Proof. The proof follows from Lemma 3.1.3.

Lemma 3.1.5. Let µ be a fuzzy subgroup of a group G. Then the following assertions hold. (1) For all x, y ∈ G, µ(x) = µ(y) implies µ(xy) = µ(x) ∧ µ(y). (2) For all x, y ∈ G, µ(xy −1 ) = µ(e) implies µ(x) = µ(y). Proof. (1) Suppose µ(x) > µ(y). Then µ(y) = µ(x−1 xy) ≥ µ(x−1 ) ∧ µ(xy) ≥ µ(x) ∧ µ(x) ∧ µ(y) = µ(y). Thus µ(xy) = µ(x) ∧ µ(y). A similar argument holds if µ(y) > µ(x). (2) Suppose µ(x) = µ(y). Then µ(x) = µ(y −1 ). Hence by (1), µ(e) = µ(xy −1 ) = µ(x) ∧ µ(y −1 ). Since µ(e) ≥ µ(z) for all z ∈ G, µ(x) = µ(y −1 ) = µ(y), a contradiction.

3.1 Commutative Fuzzy Subsets and Nilpotent Fuzzy Subgroups

Lemma 3.1.6. Let µ be a fuzzy subgroup of a group G. G | µ(xyx−1 y −1 ) = µ(e) for all y ∈ G}. Then T = Z(µ).

63

Let T = {x ∈

Proof. Let x ∈ T. Then ∀y, z ∈ G, µ((xyz)(yxz)−1 ) = µ(xyzz −1 x−1 y −1 ) = µ(xyx−1 y −1 ) = µ(e). By Lemma 3.1.5 (2), µ(xyz) = µ(yxz) for all y, z ∈ G and so x ∈ Z(µ). Therefore, T ⊆ Z(µ). Conversely, if x ∈ Z(µ), then µ(xyx−1 y −1 ) = µ(e) for all y ∈ G by Lemma 3.1.3. Thus x ∈ T. Hence Z(µ) ⊆ T. Proposition 3.1.7. Let µ be a fuzzy subgroup of a group G. Then µ(xyx−1 y −1 ) = µ(e) ∀x, y ∈ G if and only if µ is commutative in G. Proof. The proof follows from Lemma 3.1.6.

The next result is an analog of the standard result that the center Z(G) of a group G is normal in G. Proposition 3.1.8. Let µ be a fuzzy subset of a semigroup G. If Z(µ) (resp. N (µ)) is nonempty, then Z(µ) (resp. N (µ)) is a subsemigroup of G. Moreover, if G is a group, then Z(µ) is a normal subgroup of G and N (µ) is a subgroup of G. Proof. Let x1 , x2 ∈ Z(µ). Then for all y, z ∈ G, we have µ((x1 x2 )yz) = µ(y(x1 x2 )z) by Lemma 3.1.3 and clearly µ((x1 x2 )y) = µ(y(x1 x2 )). Hence x1 x2 ∈ Z(µ). Thus Z(µ) is a sub-semigroup of G if Z(µ) is nonempty. Suppose G is a group. Then Z(µ) is nonempty since e ∈ Z(µ). If x ∈ Z(µ), then µ(x−1 yz) = µ(x−1 yx−1 xz) = µ(xx−1 yx−1 z) = µ(yx−1 z) for all y, z ∈ G and so x−1 ∈ Z(µ). Hence Z(µ) is a subgroup of G by the ﬁrst part of the proof. Next let x ∈ Z(µ) and g ∈ G. Then by Lemma 3.1.3, µ((g −1 xg)yz) = µ(xg −1 gyz) = µ(xyz) = µ(xyg −1 gz) = µ(yg −1 xgz) = µ(y(g −1 xg)z) for all y, z ∈ G and so g −1 xg ∈ Z(µ). Thus Z(µ) is a normal subgroup of G if G is a group. The argument that N (µ) is a sub-semigroup of G is similar to that for Z(G). By Theorem 1.3.7, N (µ) is a subgroup of G when G is a group.

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We now consider homomorphic images and homomorphic preimages of the centralizer and the normalizer of a fuzzy subset. Proposition 3.1.9. Let f be a homomorphism of a semigroup G onto a semigroup H. Let µ and ν be fuzzy subsets of G and H, respectively. (1) If G and H are groups, then f (Z(µ)) ⊆ Z(f (µ)) and f (N (µ)) ⊆ N (f (µ)). (2) f −1 (Z(ν)) = Z(f −1 (ν)) and f −1 (N (ν)) = N (f −1 (ν)). Proof. (1) Let x ∈ f (Z(µ)). Then there exists u ∈ Z(µ) such that f (u) = x. For all y, z ∈ H, (f (µ))(xyz) = ∨{µ(a) | f (a) = xyz} = ∨{µ(uvwk) | k ∈ ker(f )} = ∨{µ(vuwk) | k ∈ ker(f )} = ∨{µ(b) | f (b) = yxz} = (f (µ))(yxz), where v, w ∈ G are such that f (v) = y and f (w) = z. Thus x ∈ Z(f (µ)). Hence f (Z(µ)) ⊆ Z(f (µ)). In a similar manner, we have that f (N (µ)) ⊆ N (f (µ)). (2) Let x ∈ f −1 (Z(ν)). Then for all y, z ∈ G, (f −1 (ν))(xyz) = ν(f (xyz)) = ν(f (x)f (y)f (z)) = ν(f (y)f (x)f (z)) = ν(f (yxz)) = (f −1 (ν))(yxz). Similarly, (f −1 (ν))(xy) = (f −1 (ν))(yx). Thus x ∈ Z(f −1 (ν)). Hence f −1 (Z(ν)) ⊆ Z(f −1 (ν)). Let x ∈ Z(f −1 (ν)) and f (x) = u. Then for all v, w ∈ H, ν(uvw) = ν(f (x)f (y)f (z)) = ν(f (xyz)) = (f −1 (ν))(xyz) = (f −1 (ν))(yxz) = ν(f (yxz)) = ν(f (y)f (x)f (z)) = ν(vuw), where y, z ∈ G are such that f (y) = v and f (z) = w. Similarly, ν(uv) = ν(vu). Thus u ∈ Z(ν), i.e., x ∈ f −1 (Z(ν)). Hence Z(f −1 (ν)) ⊆ f −1 (Z(ν)). Thus f −1 (Z(ν)) = Z(f −1 (ν)). That f −1 (N (ν)) = N (f −1 (ν)) holds similarly.

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The next result now follows: Proposition 3.1.10. Let f be a homomorphism of a semigroup G onto a semigroup H. Let µ and ν be fuzzy subsets of G and H, respectively. Then the following assertions hold. (1) If µ is commutative (normal) in G, then f (µ) is commutative (normal) in H, where G and H are groups. (2) ν is commutative (normal) in H if and only if f −1 (ν) is commutative (normal) in G. In Proposition 3.1.9(1), f (Z(µ)) = Z(f (µ)) and f (N (µ)) = N (f (µ)) in general even if µ is a fuzzy subgroup. Hence the converse of Proposition 3.1.10(1) does not hold in general, as the following example shows. Example 3.1.11. Let µ be the fuzzy subgroup of the dihedral group, D4 = a, b | a4 = e = b2 , ba = a3 b deﬁned as follows: ∀x ∈ D4 , t0 if x ∈ b µ(x) = t1 otherwise, where t0 > t1 . Let f be the natural homomorphism of D4 onto D4 /D4 , where D4 = {e, a2 } is the commutator subgroup of D4 . We have µ(a(ab)) = µ(a2 b) = t1 = t0 = µ(b) = µ((ab)a) and µ(a3 (a3 b)) = µ(a2 b) = t1 = t0 = / N (µ) and a, a3 ∈ / Z(µ). Thus f (a) ∈ / µ(b) = µ((a3 b)a3 ). Hence a, a3 ∈ f (N (µ)) and f (a) ∈ / f (Z(µ)) since f −1 (f (a)) = {a, a3 }. However, D4 /D4 is Abelian. Hence N (f (µ)) = D4 /D4 = Z(f (µ)). Thus f (N (µ)) = N (f (µ)) and f (Z(µ)) = Z(f (µ)). We now consider the centralizer and the normalizer of the intersection of fuzzy subsets. Proposition 3.1.12. (1) Let µ and ν be fuzzy subsets of a semigroup G. Then Z(µ) ∩ Z(ν) ⊆ Z(µ ∩ ν) and N (µ) ∩ N (ν) ⊆ N (µ ∩ ν). In particular, if µ and ν are commutative (invariant), then (µ ∩ ν) is commutative (invariant). (2) Let µ and ν be fuzzy subgroups of a group G such that µ(e) = ν(e). Then Z(µ) ∩ Z(ν) = Z(µ ∩ ν). In particular, µ and ν are commutative in G if and only if µ ∩ ν is commutative in G. Proof. (1) The desired result follows easily. (2) By Lemma 3.1.6, x ∈ Z(µ ∩ ν) ⇔ (µ ∩ ν)(e) = (µ ∩ ν)(xyx−1 y −1 )

for all y ∈ G −1 −1

⇔ µ(e) = ν(e) = (µ ∩ ν)(e) = µ(xyx y ) ∧ ν(xyx−1 y −1 ) for all y ∈ G ⇔ µ(xyx−1 y −1 ) = µ(e) and ν(xyx−1 y −1 ) = ν(e) for all y ∈ G ⇔ x ∈ Z(µ) and x ∈ Z(ν) ⇔ x ∈ Z(µ) ∩ Z(ν). Thus Z(µ) ∩ Z(ν) = Z(µ ∩ ν).

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We now study the centralizer and the normalizer of the sup-min product of two fuzzy subsets. Let µ and ν be fuzzy subsets of a groupoid G. Recall that the product µ ◦ ν is the fuzzy subset of G deﬁned by ∀x ∈ G, (µ ◦ ν)(x) = ∨{µ(a) ∧ ν(b) | x = ab, a, b ∈ G}. The following result is a consequence of Theorem 1.3.1. Lemma 3.1.13. Let µ be a fuzzy subset of a group G. Let S = {x ∈ G | µ(x−1 yx) = µ(y) for all y ∈ G} and T = {x ∈ G | µ(xyx−1 ) = µ(y) for all y ∈ G}. Then S = N (µ) = T. Proposition 3.1.14. Let µ and ν be fuzzy subsets of a group G. Z(µ)Z(ν) ⊆ Z(µ ◦ ν) and N (µ) ∩ N (ν) ⊆ N (µ ◦ ν).

Then

Proof. Let x1 ∈ Z(µ) and x2 ∈ Z(ν). Then for all y, z ∈ G, (µ ◦ ν)((x1 x2 )yz) = ∨{µ(a) ∧ ν(b) | x1 x2 yz = ab, a, b ∈ G} = ∨{µ(x1 x2 yzb−1 ) ∧ ν(b) | b ∈ G} = ∨{µ(x2 yx1 zb−1 ) ∧ ν(b) | b ∈ G} = ∨{µ(a) ∧ ν(b) | x2 yx1 z = ab, a, b ∈ G} = ∨{µ(a) ∧ ν(a−1 x2 yx1 z) | a ∈ G} = ∨{µ(a) ∧ ν(a−1 yx1 x2 z) | a ∈ G} = ∨{µ(a) ∧ ν(b) | yx1 x2 z = ab, a, b ∈ G} = (µ ◦ ν)(y(x1 x2 )z) by Lemma 3.1.3. Similarly, (µ ◦ ν)((x1 x2 )y) = (µ ◦ ν)(y(x1 x2 )). Hence x1 x2 ∈ Z(µ ◦ ν). Thus Z(µ)Z(ν) ⊆ Z(µ ◦ ν). Let x ∈ N (µ) ∩ N (ν). Then for all y ∈ G, (µ ◦ ν)(y) = ∨{µ(a) ∧ ν(b) | y = ab, a, b ∈ G} = ∨{µ(x−1 ax) ∧ ν(x−1 bx) | y = ab, a, b ∈ G} ≤ ∨{µ(c) ∧ ν(d) | x−1 yx = cd, c, d ∈ G} = (µ ◦ ν)(x−1 yx) by Lemma 3.1.13, where the inequality holds since y = ab ⇒ x−1 abx = cd ⇔ ab = xcdx−1 = (xcx−1 )(xdx−1 ) and since a = xcx−1 , b = xdx−1 implies x−1 ax = c, x−1 bx = d. Hence (µ ◦ ν)(x−1 yx) ≤ (µ ◦ ν)(x(x−1 yx)x−1 ) = (µ◦ν)(y) since x−1 ∈ N (µ)∩N (ν) by Proposition 3.1.8. Thus (µ◦ν)(x−1 yx) = (µ ◦ ν)(y) for all y ∈ G. Therefore, x ∈ N (µ ◦ ν) by Lemma 3.1.13. Hence N (µ) ∩ N (ν) ⊆ N (µ ◦ ν). Corollary 3.1.15. Let µ and ν be fuzzy subsets of a group G. (1) If either µ or ν is commutative in G, then µ ◦ ν is commutative in G. (2) If µ and ν are normal in G, then µ ◦ ν is normal in G.

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Proof. (1) Suppose µ is commutative in G. Then G = GZ(ν) = Z(µ)Z(ν) ⊆ Z(µ) ◦ Z(ν) ⊆ G. (2) Suppose µ is normal in G. Then G = GN (ν) = N (µ)N (ν) ⊆ N (µ) ◦ N (ν) ⊆ G. In Proposition 3.1.14, it is not the case that N (µ) ∩ N (ν) = N (µ ◦ ν), in general, and the converse of Corollary 3.1.15(1) and (2) do not hold, even if µ and ν are fuzzy subgroups, as the next example demonstrates. Example 3.1.16. Let µ and ν be the fuzzy subgroups of the dihedral group D3 = a, b | a3 = e = b2 , ba = a2 b deﬁned as follows: ∀x ∈ D3 , t0 t0 if x ∈ ab if x ∈ a µ(x) = and ν(x) = otherwise t1 t1 otherwise, where t0 > t1 . Since a is a normal subgroup of D3 , the level sets of µ are normal subgroups of D3 . Thus N (µ) = D3 = N (µ ◦ ν). (Note µ ◦ ν is a constant on D3 .) We next show that N (ν) = ab . It follows that ab = {e, ab}. Let y ∈ D3 . Then aby = e ⇒ y = ab ⇒ yab = e ∈ ab. Also, aby = ab ⇒ y = e ⇒ yab = ab ∈ ab. Now aby ∈ / ab ⇔ y ∈ / ab ⇔ yab ∈ / ab. Hence ν(yab) = ν(aby)∀y ∈ D3 .Thus it follows that ab ⊆ N (ν). Now / N (ν). Also ν((a2 b)a2 ) = ν(baa2 ) = ν(ab) = ν(a2 b) = ν(ba). Hence a, b ∈ 2 2 2 2 / N (ν). Therefore, N (ν) = ab. ν(b) = ν(ab) = ν(a (a b)). Thus a , a b ∈ Hence µ and µ ◦ ν are normal in D3 . However, ν is not normal in D3 and N (µ) ∩ N (ν) = N (µ ◦ ν). Example 3.1.17. Let G and µ be as deﬁned in Example 3.1.2. Let ν be the fuzzy subset of G deﬁned as follows: ∀x ∈ G,   t0 if x ∈ a ⊗ {e} ν(x) = t1 if x ∈ ({e} ⊗ a)\(a ⊗ {e})  t2 otherwise, where t0 > t1 > t2 . Since µ ◦ ν(ai , aj ) = µ ◦ ν((e, aj )(ai , e)), it follows that µ ◦ ν(x, y) = t0 ∀(x, y) ∈ a ⊗ a and that µ ◦ ν(x, y) = t2 ∀(x, y) ∈ D3 ⊗ D3 \(a ⊗ a). Since a is a normal subgroup of D3 , it follows that a ⊗ a is a normal subgroup of D3 ⊗ D3 = G.Thus the level sets of µ ◦ ν are normal subgroups of G. Hence µ ◦ ν is a normal fuzzy subgroup of G. Thus N (µ ◦ ν) = G. Now for all x, y, z ∈ D3 , xyz and yxz have the same number of b’s (modulo 2) in their expressions. Therefore, it follows that Z(µ ◦ ν) = G. Proposition 3.1.18. Let µ and ν be fuzzy subgroups of a group G such that µ ⊆ ν and µ(e) = ν(e). Then Z(µ) ⊆ Z(ν). In particular, if ν is commutative in G, then µ is commutative in G. Proof. Application of Lemma 3.1.6 yields the desired result.

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We next introduce the notions of the ascending central series of a fuzzy subgroup and a nilpotent fuzzy subgroup of a group. These are generalizations of the notions of the ascending central series of a group and a nilpotent group, respectively. Let µ be a fuzzy subgroup of a group G. Let Z 0 (µ) = {e} and π0 be the natural homomorphism of G onto G/Z 0 (µ). Suppose that Z i (µ) has been deﬁned and that Z i (µ) is a normal subgroup of G for i ∈ N∪{0}. Let πi be the natural homomorphism of G onto G/Z i (µ). Deﬁne Z i+1 (µ) = πi−1 (Z(πi (µ))). Then Z i+1 (µ) ⊇ Ker(πi ) = Z i (µ) for i = 0, 1, .... The normality of Z i+1 (µ) in G follows by the correspondence theorem and Proposition 3.1.8. Deﬁnition 3.1.19. Let µ be a fuzzy subgroup of a group G. The ascending central series of µ is deﬁned to be the ascending chain of normal subgroups of G, Z 0 (µ) ⊆ Z 1 (µ) ⊆ .... Deﬁnition 3.1.20. A fuzzy subgroup µ of a group G is called nilpotent if there exists a nonnegative integer m such that Z m (µ) = G. The smallest such integer m is called the class of µ. Lemma 3.1.21. Let µ be a fuzzy subgroup of a group G. xyx−1 y −1 ∈ Z i−1 (µ) for all y ∈ G, then x ∈ Z i (µ).

Let i ∈ N. If

Proof. Suppose that xyx−1 y −1 ∈ Z i−1 (µ) (= Ker(πi−1 )) for all y ∈ G. Then clearly e ∈ (πi−1 )−1 (πi−1 (xyx−1 y −1 )). Hence (πi−1 (µ))(πi−1 (xyx−1 y −1 )) = µ(e) = (πi−1 (µ))(e) for all y ∈ G. Thus πi−1 (x) ∈ Z(πi−1 (µ)) by Lemma 3.1.6 3.1.6. Hence x ∈ Z i (µ). We denote the ascending central series of a group G by {e} = Z 0 (G) ⊆ Z 1 (G) ⊆ .... Proposition 3.1.22. Let µ be a fuzzy subgroup of a group G. Then Z i (G) ⊆ Z i (µ) for all nonnegative integers i. Proof. We prove the result by induction on i. If i = 0 or 1, then the result is immediate. Assume Z i−1 (G) ⊆ Z i−1 (µ) for i > 1, the induction hypothesis. Let x ∈ Z i (G). Then xyx−1 y −1 ∈ Z i−1 (G) for all y ∈ G. Hence xyx−1 y −1 ∈ Z i−1 (µ) for all y ∈ G by the induction hypothesis. Thus x ∈ Z i (µ) by Lemma 3.1.21. Hence the desired result holds. Corollary 3.1.23. Let µ be a fuzzy subgroup of a group G. If G is nilpotent of class m, then µ is nilpotent of class n for some nonnegative integer n ≤ m. The converse of Corollary 3.1.23 is not true in general, as can be seen by the following example.

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Example 3.1.24. Let µ be the fuzzy subgroup of the dihedral group D3 = a, b | a3 = e = b2 , ba = a2 b deﬁned as follows: ∀x ∈ D3 , t0 if x ∈ a µ(x) = t1 otherwise, where t0 > t1 . Now D3 is not nilpotent. We show that µ is nilpotent. It follows easily that (D3 \a)(D3 \a) = a, a(D3 \a) = D3 \a, (D3 \a)a = D3 \a, aa = a. Hence xy ∈ a ⇔ yx ∈ a ∀x, y ∈ D3 . Thus Z(µ) = D3 . The following proposition and theorem show that the nilpotence of a group can be completely characterized by the nilpotence of the fuzzy subgroups of the group. Proposition 3.1.25. Let G be a group. Then there exists a nontrivial fuzzy subgroup µ of G such that Z i (G) = Z i (µ) for every nonnegative integer i. Proof. Let µ be the fuzzy  1 1 µ(x) = i+1  0

subgroup of G deﬁned as follows: ∀x ∈ G, if x ∈ Z 0 (G), if x ∈ Z i (G)\Z i−1 (G) for i = 1, 2, ..., otherwise.

We now show by induction on i that Z i (µ) ⊆ Z i (G) for every nonnegative integer i. If i = 0, then the result is immediate. Let i > 0 and assume Z i−1 (µ) ⊆ Z i−1 (G), the induction hypothesis. Then since µ has the sup property, x ∈ Z i (µ) ⇒ πi−1 (x) ∈ Z(πi−1 (µ)) ⇒ (πi−1 (µ))(πi−1 (xyx−1 y −1 )) = (πi−1 (µ))(πi−1 (e)) = µ(e) for all y ∈ G ⇒ ∨{µ(xyx−1 y −1 k) | k ∈ Z i−1 (µ)} = µ(e) for all y ∈ G ⇒ µ(xyx−1 y −1 k) = µ(e) for all y ∈ G and for some k ∈ Z i−1 (µ) ⇒ µ(xyx−1 y −1 k) = µ(e) for all y ∈ G and for some k ∈ Z i−1 (G) ⇒ µ(xyx−1 y −1 ) = µ(k −1 ) = µ(k) for all y ∈ G and for some k ∈ Z i−1 (G) 1 for all y ∈ G ⇒ µ(xyx−1 y −1 ) ≥ i ⇒ xyx−1 y −1 ∈ Z i−1 (G) for all y ∈ G ⇒ x ∈ Z i (G)

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by Lemma 3.1.6 and the induction hypothesis. Thus Z i (µ) ⊆ Z i (G). By Proposition 3.1.22, Z i (G) ⊆ Z i (µ). Hence µ is a desired fuzzy subgroup of G. Theorem 3.1.26. Let G be a group. Then G is nilpotent if and only if all fuzzy subgroups of G are nilpotent. Proof. The proof follows from Corollary 3.1.23 and Proposition 3.1.25.

We now consider homomorphic images and homomorphic pre-images of nilpotent fuzzy subgroups. Proposition 3.1.27. Let f be a homomorphism of a group G onto a group H. Let µ and ν be fuzzy subgroups of G and H, respectively. (1) If µ is nilpotent of class m, then f (µ) is nilpotent of class n for some nonnegative integer n ≤ m. (2) ν is nilpotent of class m if and only if f −1 (ν) is nilpotent of class m. Proof. (1) We show by induction on i that f (Z i (µ)) ⊆ Z i (f (µ)) for every nonnegative integer i. If i = 0 or 1, then the result is clear by Proposition 3.1.9(1). Let i > 1 and assume f (Z i−1 (µ)) ⊆ Z i−1 (f (µ)), the induction hypothesis. Let h ∈ f (Z i (µ)). Then there exists x ∈ Z i (µ) such that f (x) = h and for all y ∈ G, ∨{µ(xyx−1 y −1 k) | k ∈ Z i−1 (µ)} = (πi−1 (µ))(πi−1 (xyx−1 y −1 )) = µ(e) by Lemma 3.1.21, where πi−1 is the natural homomorphism of G onto G/Z i−1 (µ). For all y ∈ G, (πi−1 (f (µ)))(πi−1 (f (x)f (y)f (x)−1 f (y)−1 ))

= ∨{f (µ)(f (x)f (y)f (x)−1 f (y)−1 f (k)) | f (k) ∈ Z i−1 (f (µ))} ≥ ∨{(f (µ))(f (x)f (y)f (x)−1 f (y −1 )f (k)) | f (k) ∈ f (Z i−1 (µ)} ≥ ∨{(f (µ))(f (xyx−1 y −1 k) | k ∈ Z i−1 (µ)} = ∨{∨{µ(xyx−1 y −1 kg) | g ∈ ker(f )} | k ∈ Z i−1 (µ)} ≥ ∨{µ(xyx−1 y −1 k) | k ∈ Z i−1 (µ)} by the induction hypothesis, where πi−1 is the natural homomorphism of i−1 (f (x)f (y)f (x)−1 f (y)−1 )) = H onto H/Z (f (µ)). Thus (πi−1 (f (µ)))(πi−1 µ(e) = (f (µ))(e) for all y ∈ G. Therefore, h = f (x) ∈ Z i (f (µ)) by Lemma 3.1.6. Hence f (Z i (µ)) ⊆ Z i (f (µ)). (2) We now show by induction on i that Z i (f −1 (ν)) = f −1 (Z i (ν)) for every positive integer i. If i = 1, the result follows by Proposition 3.1.9(2). Let i > 1 and assume Z i−1 (f −1 (ν)) = f −1 (Z i−1 (ν)), the induction hypothesis. Then for all y ∈ G,

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71

(πi−1 (f −1 (ν)))(πi−1 (xyx−1 y −1 ))

= ∨{(f −1 (ν))(xyx−1 y −1 k) | k ∈ Z i−1 (f −1 (ν))} = ∨{ν(f (xyx−1 y −1 k)) | k ∈ Z i−1 (f −1 (ν))} = ∨{ν(f (xyx−1 y −1 )f (k)) | k ∈ f −1 (Z i−1 (ν))} = ∨{ν(f (xyx−1 y −1 )h) | h ∈ Z i−1 (ν)} = (πi−1 (ν))(πi−1 (f (xyx−1 y −1 ))) by the induction hypothesis, where πi−1 (resp. πi−1 ) is the natural homomori−1 phism from H (resp. G) onto H/Z (ν) (resp. G/Z i−1 (f −1 (ν))). Thus

x ∈ Z i (f −1 (ν)) ⇔ πi−1 (x) ∈ Z(πi−1 (f −1 (ν))) ⇔ (πi−1 (f −1 (ν)))(πi−1 (xyx−1 y −1 )) = ν(e)

⇔ (πi−1 (ν))(πi−1 (f (xyx−1 y −1 ))) = ν(e) −1 ⇔ f (x) ∈ πi−1 (Z(πi−1 (ν)))

for all y ∈ G for all y ∈ G

⇔ f (x) ∈ Z i (ν) ⇔ x ∈ f −1 (Z i (ν)) by Lemma 3.1.6. Hence Z i (f −1 (ν)) = f −1 (Z i (ν)).

Corollary 3.1.28. Let µ be a fuzzy subgroup of a group G such that µ∗ is normal in G. If G/µ∗ is nilpotent of class m, then µ is nilpotent of class n for some nonnegative integer n ≤ m. Proof. Suppose G/µ∗ is nilpotent of class m. Let f be the natural homomorphism of G onto G/µ∗ . Then f (µ) is nilpotent of class n for some nonnegative integer n ≤ m by Corollary 3.1.23. Now f −1 (f (µ))(x) = f (µ)(f (x)) = ∨{µ(z) | f (z) = f (x), z ∈ G}. Now ∀x, z ∈ G, f (z) = f (x) ⇔ f (z −1 x) = e ⇔ z −1 x ∈ Ker f = µ∗ ⇔ µ(z −1 x) = µ(e) ⇒ µ(z) = µ(x). Hence it follows that f −1 (f (µ)) = µ. Thus µ is nilpotent of class n by Proposition 3.1.27(2). The converse of Proposition 3.1.27(1) does not hold in general, as the following example shows. Example 3.1.29. Let µ be the fuzzy subset of the dihedral group D3 = a, b | a3 = e = b2 , ba = a2 b deﬁned as follows: ∀x ∈ D3 , µ(x) =

t0 t1

if x = e, otherwise,

/ Z(µ) since where t0 > t1 . Then µ is a fuzzy subgroup of D3 . Now a ∈ µ(a(ba)b) = µ((ab)2 ) = µ(e) = t0 = t1 = µ(a) = µ(ba2 b) = µ((ba)ab). Thus Z(µ) = {e} since the only normal subgroups of D3 are {e}, a , and D3

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and Z(µ) is normal in D3 by Proposition 3.1.8. Hence Z i (µ) = {e} for all nonnegative integers i. Thus µ is not nilpotent. Let f be the natural homomorphism of D3 onto D3 /a. Then Z(f (µ)) = D3 /a since D3 /a is Abelian. Hence f (µ) is nilpotent. We now give an analog of Proposition 3.1.18. Proposition 3.1.30. Let µ and ν be fuzzy subgroups of a group G such that µ ⊆ ν and µ(e) = ν(e). If µ is nilpotent of class m, then ν is nilpotent of class n for some nonnegative integer n ≤ m. Proof. We integers i. Let i > 1 x ∈ Z i (µ).

show by induction on i that Z i (µ) ⊆ Z i (ν) for all nonnegative If i = 0 or 1, then the result follows by Proposition 3.1.18 3.1.18. and assume Z i−1 (µ) ⊆ Z i−1 (ν), the induction hypothesis. Let Then for all y ∈ G,

(πi−1 (ν))(πi−1 (xyx−1 y −1 )) = ∨{ν(xyx−1 y −1 k) | k ∈ Z i−1 (ν)}

≥ ∨{ν(xyx−1 y −1 k) | k ∈ Z i−1 (µ) ≥ ∨{µ(xyx−1 y −1 k) | k ∈ Z i−1 (µ)} = (πi−1 (µ))(πi−1 (xyx−1 y −1 ) = µ(e) by Lemma 3.1.6 and the induction hypothesis. Hence x ∈ Z i (ν) by Lemma 3.1.6. Thus Z i (µ) ⊆ Z i (ν).

3.2 Nilpotent Fuzzy Subgroups The notion of a solvable fuzzy subgroup was introduced in [15]. In the previous section, the notion of a nilpotent fuzzy subgroup was introduced. In the previous section, an ascending series of subgroups of the underlying group was attached to a fuzzy subgroup to deﬁne nilpotency of the fuzzy subgroup. We deﬁne the commutator of a pair of fuzzy subsets of a group and use this technique to generate the descending central chain of fuzzy subgroups of a given fuzzy subgroup. We then propose a deﬁnition of a nilpotent fuzzy subgroup through its descending central chain. It is known that every Abelian group is nilpotent. If we accept the deﬁnition of an Abelian fuzzy group as proposed in [14], where a fuzzy group is called Abelian if its support is Abelian, then one can verify, as is done in Theorem 3.2.25, that Abelian fuzzy subgroups are nilpotent. The power set P(S) of S can be embedded in FP(S) as a sublattice under the identiﬁcation map A → 1A , where 1A is the characteristic function of the subset A of S. For arbitrary λ in FP(G), recall that the least fuzzy subgroup of G containing λ is denoted by λ. Then for all x ∈ G, λ(x) =

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73

∨{σ(x1 ) ∧ σ(x2 ) ∧ . . . ∧ σ(xn ) | x = x1 x2 ., ..xn , xi ∈ G, i = 1, 2, ...n; n ∈ N}, where σ(y) = λ(y) ∨ λ(y −1 ), y ∈ G. The set F(G) of all fuzzy subgroups of G is a complete lattice in which the meet and the join of λ and µ are, respectively, λ ∩ µ and λ ∪ µ. The lattice F(G) contains the set of all subgroups of G as a sublattice. For fuzzy subsets λ and µ of a group G, recall that the fuzzy subset λ ◦ µ of G is deﬁned by λ ◦ µ(x) = ∨{λ(a) ∧ µ(b) | x = ab, a, b ∈ G} for all x ∈ G. A nonempty collection D of endomorphisms of a group G is called an operator domain on G. A fuzzy subset λ of G is called admissible under D or simply D admissible if for every f ∈ D, f (λ) ⊆ λ. It follows that f (λ) ⊆ λ if and only if λ ⊆ f −1 (λ). A fuzzy subgroup λ of G is normal (characteristic, fully invariant) if λ is admissible under all inner automorphisms (automorphisms, endomorphisms) of G. The sup-min product of two D-admissible fuzzy subsets is D-admissible. Moreover, if λ is a D-admissible fuzzy subset of G, then so is λ; see [1]. Recall that if λ is a normal fuzzy subgroup and µ any fuzzy subgroup of G, then λ ◦ µ is a fuzzy subgroup of G. In a group G, the commutator of two elements a and b of G is the element [a, b] = a−1 b−1 ab of G. If A, B ⊆ G, then the commutator subgroup of A and B is the subgroup [A, B] of G generated by {[a, b] | a ∈ A, b ∈ B}. For all x, y ∈ G, let xy denote y −1 xy. Then the following properties hold: [a, b]−1 = [b, a], [ab, c] = [a, c]b [b, c], and f ([a, b]) = [f (a), f (b)],where a, b, c ∈ G and f is an endomorphism of G. Let G be a group with identity e and let H be a subgroup of G. Let Zn (H) = [Zn−1 (H), H]∀n ∈ N,where Z0 (H) = H. Then Zn (H) is a subgroup of G ∀n ∈ N. The chain H = Z0 (H) ⊇ Z1 (H) ⊇ ... ⊇ Zn (H) ⊇ ... is called the descending central chain of the subgroup H. The subgroup H is called nilpotent if Zn (H) = {e} for some n ≥ 0. In fact, H is called nilpotent of class c if c is the smallest nonnegative integer such that Zc (H) = {e}. The group G is said to be nilpotent if it is nilpotent as a subgroup of itself. Deﬁnition 3.2.1. Let λ and µ be fuzzy subsets of G. Let (λ, µ) be the fuzzy subset of G deﬁned as follows: ∀x ∈ G, ∨{λ(a) ∧ µ(b) | x = [a, b], a, b ∈ G} if x is a commutator of G (λ, µ)(x) = 0 otherwise. The commutator of λ and µ is the fuzzy subgroup [λ, µ] of G generated by (λ, µ). The following theorem shows that the notion of a commutator fuzzy subgroup contains the corresponding notion for crisp subgroups. Theorem 3.2.2. Let A and B be subsets of G. Then [1A , 1B ] = 1[A,B] .

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Proof. We ﬁrst show that for any subset T of G, 1T = 1T , where T is the subgroup of G generated by T. Let σ(x) = 1T (x) ∨ 1T (x−1 )∀x ∈ G. Then 1T (x) = ∨{σ(a1 ) ∧ . . . ∧ σ(an ) | x = a1 ...an , ai ∈ G, i = 1, ..., n; n ∈ N}, x ∈ G. Suppose x ∈ T . Then there exists a1 , . . . , an ∈ G such that x = a1 . . . an and for each i, either ai ∈ T or a−1 ∈ T. We then get σ (ai ) = 1 for each i i and 1 ≥ 1T (x) ≥ {σ(a1 ) ∧ . . . ∧ σ(an ) = 1. / T . We show that This yields 1T (x) = 1 = 1T (x). Suppose x ∈ 1T (x) = 0. If 1T (x) > 0, then there is a decomposition x = a1 . . . an of x in G such that 0 < σ(a1 ) ∧ . . . ∧ σ(an ). Thus for all i, σ(ai ) = 1T (ai ) ∨ −1 ∈ T for all i. Consequently, x ∈ T ,a 1T (a−1 i ) = 1. Hence either ai ∈ T or ai contradiction. Hence 1T (x) = 0 = 1T (x). Now let A, B be any two subsets of G. Clearly, for all x ∈ G, (1A , 1B )(x) = 1 if x ∈ {[a, b] | a ∈ A, b ∈ B} and 0 otherwise. Thus (1A , 1B ) = 1{[a,b] | a∈A,b∈B} . By the above argument, we have that [1A , 1B ] = 1[A,B] . The following theorem is concerned with the support and the tip of the commutator fuzzy subgroup of a pair of fuzzy subsets of G. Recall that µ∗ denotes the support of a fuzzy subset µ of G. Theorem 3.2.3. Let λ and µ be fuzzy subsets of G. Then the following assertions hold. (1) [λ, µ]∗ = [λ∗ , µ∗ ]. (2) If r, s are the tips of λ, µ, respectively, then t = r ∧ s is the tip of [λ, µ]. Proof. (1) Let x ∈ (λ, µ)∗ . Then (λ, µ)(x) > 0 and so there exist a, b ∈ G such that x = [a, b] and λ(a) ∧ µ(b) > 0. Consequently, a ∈ λ∗ , b ∈ µ∗ and x ∈ {[l, m] | l ∈ λ∗ , m ∈ µ∗ }. Thus supp((λ, µ)) ⊆ {[l, m] | l ∈ λ∗ , m ∈ µ∗ }. Suppose x ∈ {[l, m] | l ∈ λ∗ , m ∈ µ∗ }. Then there exists l ∈ λ∗ , m ∈ µ∗ such that x = [l, m] and 0 < λ(l) ∧ µ(m) ≤ (λ, µ)([l, m]). Hence {[l, m] | l ∈ λ∗ , m ∈ µ∗ } ⊆ (λ, µ)∗ and so {[l, m] | l ∈ λ∗ , m ∈ µ∗ } = (λ, µ)∗ ⊆ [λ, µ]∗ . Since [λ, µ]∗ is a subgroup of G and [λ∗ , µ∗ ] is the smallest subgroup of G containing {[l, m] | l ∈ λ∗ , m ∈ µ∗ }, we have [λ∗ , µ∗ ] ⊆ [λ, µ]∗ . On the other hand, (λ, µ)∗ = {[l, m] | l ∈ λ∗ , m ∈ µ∗ } ⊆ [λ∗ , µ∗ ] implies (λ, µ) ⊆ 1[λ∗ ,µ∗ ] . Since 1[λ∗ ,µ∗ ] is a fuzzy subgroup of G and [λ, µ] is the smallest fuzzy subgroup of G containing (λ, µ), we have [λ, µ] ⊆ 1[λ∗ ,µ∗ ] . Thus [λ, µ]∗ ⊆ [λ∗ , µ∗ ]. Hence [λ, µ]∗ = [λ∗ , µ∗ ]. (2) Let u be the tip of [λ, µ]. Let x ∈ G. If x is not a commutator in G, then [λ, µ](x) = 0 ≤ t. Suppose x is a commutator in G. Then [λ, µ](x) = ∨{λ(a) ∧ µ(b) | x = [a, b], a, b ∈ G} ≤ r ∧ s = t. Thus u ≤ t. Suppose u < t. Then u < r and u < s and so there exist a, b ∈ G such that u < λ(a), u < µ(b). Hence u < λ(a) ∧ µ(b) ≤ [λ, µ]([a, b]). However, this contradicts the fact that u is the tip of [λ, µ]. Thus u = t.

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75

Theorem 3.2.4. Let λ, µ be fuzzy subsets of G. Then [λ, µ] = [µ, λ]. Proof. Let x ∈ G. We ﬁrst show that (λ, µ)(x) = (µ, λ)(x−1 ). If x is not a commutator in G, then x−1 is not a commutator and so (λ, µ)(x) = 0 = (µ, λ)(x−1 ). Suppose x = [a, b] for some a, b ∈ G. Then (λ, µ)(x) = ∨{λ(a) ∧ µ(b) | x = [a, b], a, b ∈ G} = ∨{µ(b) ∧ λ(a) | x−1 = [a, b], a, b ∈ G} = (µ, λ)(x−1 ). If σ(x) = (λ, µ)(x) ∨ (λ, µ)(x−1 ) and δ(x) = (µ, λ)(x) ∨ (µ, λ)(x−1 ), then σ = δ and hence [λ, µ] = σ = δ = [µ, λ]. The following two theorems are concerned with the admissibility of commutator fuzzy subgroups under operator domains on G. Theorem 3.2.5. If λ and µ are admissible fuzzy subsets of G under an operator domain D, then (λ, µ) and [λ, µ] are D-admissible. Proof. Since λ and µ are D-admissible, λ ⊆ f −1 (λ), µ ⊆ f −1 (µ) for all f ∈ D. Let f ∈ D and x ∈ G. If x is not a commutator in G, then (λ, µ)(x) = 0 ≤ (λ, µ)(f (x)). Suppose x = [a, b] for some a, b ∈ G. Then (λ, µ)(x) = ∨{λ(a) ∧ µ(b) | x = [a, b], a, b ∈ G} ≤ ∨{λ(f (a)) ∧ µ(f (b)) | x = [a, b], a, b ∈ G} = ∨{λ(f (a)) ∧ µ(f (b)) | f (x) = [f (a), f (b)], a, b ∈ G} ≤ ∨{λ(c) ∧ µ(d) | f (x) = [c, d], c, d ∈ G} = (λ, µ)(f (x)). Thus (λ, µ) ⊆ f −1 ((λ, µ)). Hence (λ, µ) is D-admissible. Thus (λ, µ) ⊆ f −1 ((λ, µ)) and since [λ, µ] is generated by (λ, µ), we have that (λ, µ) ⊆ f −1 ([λ, µ]). Hence [λ, µ] ⊆ f −1 ([λ, µ]). Therefore, [λ, µ] is D-admissible. Theorem 3.2.6. If λ and µ are normal (characteristic, fully invariant) fuzzy subgroups of G, then [λ, µ] is a normal (characteristic, fully invariant) fuzzy subgroup of G contained in λ ∩ µ. Proof. If λ and µ are normal (characteristic, fully invariant) fuzzy subgroups of G, it follows from Theorem 3.2.5 that [λ, µ] is a normal (characteristic, fully invariant) fuzzy subgroup of G. In each of the given cases, λ and µ are normal and thus they assume constant values on the conjugacy classes of G. Since λ ∩ µ is a fuzzy subgroup, it suﬃces to prove that (λ, µ) ⊆ λ ∩ µ. Let x ∈ G. If x is not a commutator in G, then (λ, µ)(x) = 0 ≤ (λ ∩ µ)(x). Suppose x = [a, b] for some a, b ∈ G. Then λ(x) = λ(a−1 (b−1 ab)) ≥ λ(a−1 ) ∧ λ(b−1 ab) = λ(a−1 ) ∧ λ(a) = λ(a). Similarly, µ(x) ≥ µ(b). Therefore, λ(a) ∧ µ(b) ≤ λ(x) ∧ µ(x) = (λ ∩ µ)(x). Hence (λ, µ)(x) ≤ (λ ∩ µ)(x).

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Lemma 3.2.7. If λ and µ are fuzzy subsets of G such that λ ⊆ µ, then [λ, σ] ⊆ [µ, σ] for every fuzzy subset σ of G. Proof. Since λ ⊆ µ, λ(x) ≤ µ(x) for all x ∈ G. Let σ be any fuzzy subset of G and let x ∈ G. If x is not a commutator in G, then (λ, σ)(x) = 0 = (µ, σ)(x). Suppose x is a commutator in G. Then (λ, σ)(x) = ∨{λ(a) ∧ σ(b) | x = [a, b], a, b ∈ G} ≤ ∨{µ(a) ∧ σ(b) | x = [a, b], a, b ∈ G} ⊆ (µ, σ)(x). Therefore, (λ, σ) ⊆ (µ, σ) and so [λ, σ] ⊆ [µ, σ]. Theorem 3.2.8. Let λ and µ be normal fuzzy subgroups of G and let σ be any fuzzy subgroup of G. Then [λ ◦ σ, µ] ⊆ [λ, µ] ◦ [σ, µ] with equality holding if λ(e) = σ(e). Proof. First we show that (λ ◦ σ, µ) ⊆ [λ, µ] ◦ [σ, µ]. Let x ∈ G. If x is not a commutator in G, then (λ ◦ σ, µ)(x) = 0 and the result is immediate. Suppose x is a commutator in G. Then (λ ◦ σ, µ)(x) = ∨{(λ ◦ σ)(a) ∧ µ(b) | x = [a, b] ∈ G} = ∨{∨{λ(u) ∧ σ(v) | a = uv, u, v ∈ G} ∧ µ(b) | x = [a, b], a, b ∈ G} = ∨{∨{(λ(u) ∧ σ(v)) ∧ µ(b) | a = uv, u, v ∈ G} | x = [a, b], a, b ∈ G} = ∨{∨{(λ(u)∧µ(b))∧(σ(v)∧µ(b)) | a = uv, u, v ∈ G} | x = [a, b], a, b ∈ G} = ∨{(λ(u) ∧ µ(b)) ∧ (σ(v) ∧ µ(b))|x = [uv, b], u, v ∈ G} ≤ ∨{[λ, µ]([u, b]) ∧ [σ, µ]([v, b]) | x = [uv, b], u, v ∈ G} = ∨{[λ, µ]([u, b]v ) ∧ [σ, µ]([v, b]) | x = [uv, b], u, v ∈ G} (since [λ, µ] is normal) ≤ ∨{[λ, µ](y) ∧ [σ, µ](z) | x = yz, y, z ∈ G}, (since [uv, b] = [u, b]v [v, b] and so y = [u, b]v , z = [v, b] ⇒ yz = [uv, b]) = ([λ, µ] ◦ [σ, µ])(x). Since [λ, µ] is normal in G, [λ, µ]◦[σ, µ] is a fuzzy subgroup of G. Therefore, [λ ◦ σ, µ] ⊆ [λ, µ] ◦ [σ, µ]. Finally, suppose λ(e) = σ(e). Then λ ⊆ λ ◦ σ and σ ⊆ λ ◦ σ. Thus [λ, µ] ⊆ [λ ◦ σ, µ] and [σ, µ] ⊆ [λ ◦ σ, µ] by Lemma 3.2.7. Hence [λ, µ] ◦ [σ, µ] ⊆ [λ ◦ σ, µ] since [λ ◦ σ, µ] is a fuzzy subgroup of G. Consequently, the desired equality holds. We now consider homomorphic images and preimages of commutators of fuzzy subsets of groups. Lemma 3.2.9. Let f be a homomorphism of a group G into a group K. For all fuzzy subsets λ of G, f (λ) = f (λ).

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Proof. Since λ ⊆ λ, f (λ) ⊆ f (λ). Since f (λ) is a fuzzy subgroup of K, f (λ) ⊆ f (λ). Suppose y ∈ K and y = f (x), x ∈ G. We show that λ(x) ≤ f (λ)(y). Deﬁne σ(t) = λ(t) ∨ λ(t−1 ), ∀ t ∈ G and δ(s) = f (λ)(s) ∨ f (λ)(s−1 ), ∀ s ∈ K. If x = a1 . . . an, then since λ(t) ≤ f (λ)(f (t)) for all t ∈ G, we have that σ(a1 ) ∧ . . . ∧ σ(an ) = ∧{λ(ai ) ∨ λ(a−1 i ) | 1 ≤ i ≤ n} ≤ ∧{f (λ)(f (ai )) ∨ f (λ)(f (a−1 i ))} | 1 ≤ i ≤ n} = ∧{δ(f (ai )) | 1 ≤ i ≤ n} ≤ f (λ)(f (a1 ) . . . f (an )) = f (λ)(y). Hence f (λ)(y) = ∨{λ(x) | y = f (x)} ≤ f (λ)(y). If y ∈ K is such that there is no x ∈ G such that f (x) = y, then f (λ)(y) = 0. Therefore, f (λ) ⊆ f (λ). Corollary 3.2.10. Let f be a homomorphism of a group G into a group K. If λ and µ are fuzzy subsets of G, then f ([λ, µ]) is the fuzzy subgroup of K generated by f ((λ, µ)). Proof. Since [λ, µ] is the fuzzy subgroup of G generated by the fuzzy subset (λ, µ), the result immediately follows from Lemma 3.2.9. In particular, f ([λ, µ]) = f ((λ, µ)) = f ((λ, µ)). Theorem 3.2.11. Let f be a homomorphism of a group G into a group K. Then for all fuzzy subsets λ and µ of G, [f (λ), f (µ)] = f ([λ, µ]) in F(K). Proof. We ﬁrst show that f ((λ, µ) ⊆ [f (λ), f (µ)]. Let y ∈ K. If f −1 (y) = ∅, then f ((λ, µ))(y) = 0 ≤ [f (λ), f (µ)](y). Suppose y = f (x) for some x ∈ G. Then (λ, µ)(x) = ∨{λ(a) ∧ µ(b) | x = [a, b], a, b ∈ G} ≤ ∨{f (λ)(f (a)) ∧ f (µ)(f (b)) | y = [f (a), f (b)], a, b ∈ G} ≤ ∨{f (λ)(c) ∧ f (µ)(d) | y = [c, d], c, d ∈ G} = (f (λ), f (µ))(y) ≤ [f (λ), f (µ)](y). Hence f ((λ, µ))(y) = ∨{(λ, µ)(x) | y = f (x), x ∈ G} ≤ [f (λ), f (µ)](y). We thus have that f ((λ, µ)) ⊆ [f (λ), f (µ)]. Hence by Corollary 3.2.10, f ([λ, µ]) ⊆ [f (λ), f (µ)]. We next show that (f (λ), f (µ)) ⊆ f ([λ, µ]). Let y ∈ K. If y is not a commutator in K, then (f (λ), f (µ))(y) = 0 ≤ f ([λ, µ])(y). Suppose y is a commutator in K. Then y = [u, v] for some u, v ∈ K. If either f −1 (u) = ∅ or f −1 (v) = ∅, then f (λ)(u) ∧ f (µ)(v) = 0. Otherwise, we have

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f (λ)(u) ∧ f (µ)(v) = (∨{λ(s) | u = f (s)}) ∧ (∨{µ(t) | v = f (t)}) = ∨{λ(s) ∧ µ(t) | u = f (s), v = f (t)} ≤ ∨{(λ, µ)([s, t]) | y = f ([s, t])} ≤ ∨{[λ, µ]([s, t]) | y = f ([s, t])} ≤ ∨{[λ, µ](x) | y = f (x)} = f ([λ, µ])(y). Thus (f (λ), f (µ)) ⊆ f ([λ, µ]). Hence [f (λ), f (µ)] ⊆ f ([λ, µ]).

Theorem 3.2.12. Let f be a homomorphism of a group G into a group H. For all fuzzy subsets ν and ρ of K, [f −1 (ν), f −1 (ρ)] ⊆ f −1 ([ν, ρ]) in F(G). Proof. Let x ∈ G. If x is not a commutator in G, then (f −1 (ν), f −1 (ρ))(x) = 0 ≤ f −1 ([ν, ρ])(x). Suppose now that x is a commutator in G. Then (f −1 (ν), f −1 (ρ))(x) = ∨{ν(f (a)) ∧ ρ(f (b)) | x = [a, b], a, b ∈ G} = ∨{ν(f (a)) ∧ ρ(f (b)) | f (x) = [f (a), f (b)], a, b ∈ G} ≤ ∨{ν(c) ∧ ρ(d) | f (x) = [c, d], c, d ∈ G} = (ν, ρ)(f (x)) ≤ [ν, ρ](f (x)) = f −1 ([ν, ρ])(x). Therefore, (f −1 (ν), f −1 (ρ)) ⊆ f −1 ([ν, ρ]). Since f −1 ([ν, ρ]) is a fuzzy sub group of G, we obtain [f −1 (ν), f −1 (ρ)] ⊆ f −1 ([ν, ρ]). We next deﬁne the notion of the descending central chain of a fuzzy subgroup. Let λ be a fuzzy subgroup of a group G. Deﬁne Z0 (λ) = λ and Z1 (λ) = [λ, λ]. Suppose Zn−1 (λ) has been deﬁned for n ∈ N. Deﬁne Zn (λ) = [Zn−1 (λ), λ], n ∈ N. Theorem 3.2.13. If H is a subgroup of G, then Zn (1H ) = 1Zn (H) for all n ∈ N ∪ {0}. Proof. The result follows from Theorem 3.2.2.

Suppose λ is a normal fuzzy subgroup of G. Let n ∈ N, n > 1. In view of Theorem 3.2.6, we have that Zn (λ) = [Zn−1 (λ), λ] ⊆ Zn−1 (λ) ∩ λ. Hence Zn (λ) ⊆ Zn−1 (λ). The following theorem shows that this inclusion holds without the assumption that λ is normal. Theorem 3.2.14. Let λ be a fuzzy subgroup of G. Then Zn (λ) ⊆ Zn−1 (λ) for all n ∈ N. Proof. We prove the result by induction on n. Let x ∈ G. If x is not a commutator in G, then (λ, λ)(x) = 0 ≤ λ(x). Suppose x is a commutator in G. Then (λ, λ)(x) = ∨{λ(a) ∧ λ(b) | x = [a, b], a, b ∈ G} = ∨{λ(a−1 ) ∧ λ(b−1 ) ∧ λ(a) ∧ λ(b) | x = [a, b], a, b ∈ G}

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≤ ∨{λ(a−1 b−1 ab) | x = [a, b], a, b ∈ G} = λ(x). Thus (λ, λ) ⊆ λ. Since λ is a fuzzy subgroup, we have that Z1 (λ) = [λ, λ] = [(λ, λ)] ⊆ λ = Z0 (λ). Therefore, the result holds for n = 1. Now suppose that Zk (λ) ⊆ Zk−1 (λ) for some k ≥ 1, the induction hypothesis. Then by Lemma 3.2.7, Zk+1 (λ) = [Zk (λ), λ] ⊆ [Zk−1 (λ), λ] = Zk (λ). Hence the desired result follows by induction. Deﬁnition 3.2.15. Let λ be a fuzzy subgroup of G. Then the chain λ = Z0 (λ) ⊇ Z1 (λ) ⊇ . . . ⊇ Zn (λ) ⊇ . . . of fuzzy subgroups of G is called the descending central chain of λ. Theorem 3.2.16. If λ is a normal (characteristic, fully invariant) fuzzy subgroup of G, then each every subgroup in the descending central chain of λ is normal (characteristic, fully invariant) in G. Proof. The result follows from Theorem 3.2.6.

Let µ1 , µ2 , . . . , µn be fuzzy subgroups of a group G. Deﬁne the fuzzy subgroup [µ1 , µ2 , . . . , µn ] of G recursively as follows: [µ1 , µ2 , . . . , µn ] = [[µ1 , µ2 , . . . , µn−1 ], µn ], n ∈ N and n ≥ 3. If the µi are normal fuzzy subgroups of G, then [µ1 , µ2 , . . . , µn ] is a normal fuzzy subgroup of G contained ∩ni=1 µi by Theorem 3.2.6. Theorem 3.2.17. Let λ be a normal fuzzy subgroup of G. If µ1 , . . . , µn+1 are normal fuzzy subgroups of G such that µi = λ for k + 1(k ≥ 0) distinct values of i, then [µ1 , . . . , µn+1 ] ⊆ Zk (λ). Proof. We prove the result by induction on n. For n = 0, there is nothing to prove. If λ, µ are normal fuzzy subgroups of G, then [λ, µ] = [µ, λ] ⊆ λ = Z0 (λ) and [λ, λ] = Z1 (λ). Thus the result is true for n = 1 and for all possible values of k. Assume the result is true for n − 1 and for all possible values of k, the induction hypothesis. Let µ1 , . . . , µn+1 be normal fuzzy subgroups of G such that µi = λ for k + 1 (k ≥ 0) distinct values of i. Suppose k = 0. If µn+1 = λ, then since [µ1 , . . . , µn ] is a normal fuzzy subgroup of G, we have that [µ1 , . . . , µn+1 ] = [[µ1 , . . . , µn ], λ] ⊆ λ = Z0 (λ). If µn+1 = λ, then by the induction hypothesis, [µ1 , . . . , µn ] ⊆ Z0 (λ) and thus [µ1 , . . . , µn+1 ] ⊆ [Z0 (λ), µn+1 ] ⊆ Z0 (λ). Suppose k ≥ 1. If µn+1 = λ, then by the induction hypothesis, [µ1 , . . . , µn ] ⊆ Zk (λ). Therefore, [µ1 , . . . , µn+1 ] ⊆ Zk (λ). If µn+1 = λ, then again by the induction hypothesis, [µ1 , . . . , µn ] ⊆ Zk−1 (λ). Thus [µ1 , . . . , µn+1 ] ⊆ [Zk−1 (λ), λ] = Zk (λ). Hence the result holds by induction on n. Deﬁnition 3.2.18. Let λ be a fuzzy subgroup of G. A chain λ = λ0 ⊇ λ1 ⊇ . . . ⊇ λn ⊇ . . . of fuzzy subgroups of G is called a central chain of λ if for all n ∈ N, λn (a)∧ λ(b) ≤ λn+1 ([a, b]) for all a, b ∈ G.

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In Deﬁnition 3.2.18, the condition is equivalent to saying that [λn , λ] ⊆ λn+1 for all n ≥ 0. Let t denote the tip of λ. If λn = et for some n ∈ N, then the series of fuzzy subgroups λ = λ0 ⊇ λ1 ⊇ . . . ⊇ λn = et is called a central series of λ. Deﬁnition 3.2.19. Let λ be a fuzzy subgroup of G with tip t. If the descending central chain λ = Z0 (λ) ⊇ Z1 (λ) ⊇ . . . ⊇ Zn (λ) ⊇ . . . of λ is such that Zn (λ) = et for some n ∈ N, then λ is called nilpotent. If λ is nilpotent, then λ is called nilpotent of class c if c is the smallest nonnegative integer such that Zc (λ) = et . In this case, the series λ = Z0 (λ) ⊇ Z1 (λ) ⊇ . . . ⊇ Zc (λ) = et is called the descending central series of λ. We show in Example 3.2.31 that the deﬁnition of nilpotence here is not equivalent to the one given in Section 3.1. This can be seen from Example 3.1.24. In the remainder of this section, a fuzzy subgroup of a group is nilpotent if it satisﬁes Deﬁnition 3.2.19. The concept of nilpotency for a fuzzy subgroup of a group is not considered further in this book. Theorem 3.2.20. Let H be a subgroup of G. Then H is nilpotent if and only if 1H is nilpotent. Proof. Consider the descending central chain H = Z0 (H) ⊇ Z1 (H) ⊇ . . . ⊇ Zn (H) ⊇ . . . of H. By Theorem 3.2.13, Zn (1H ) = 1Zn (H) for all n ∈ N. Therefore, Zn (1H ) = e1 if and only if Zn (H) = {e}. The desired result now follows. Example 3.2.21. We now give a nontrivial example of a nilpotent fuzzy subgroup of the group S4 , the symmetric group on {1, 2, 3, 4}. Let D4 = (24), (1234) = {(1), (12)(34), (13)(24), (14)(23), (13), (24), (1234), (1432)}. Then D4 is a dihedral subgroup of S4 with center C = {(1), (13)(24)}. Let λ be the fuzzy subset of S4 deﬁned by λ(x) = 1 if x ∈ C, λ(x) = 12 if x ∈ 1234 \C, λ(x) = 14 if x ∈ D4 \ 1234 , and λ(x) = 0 if x ∈ S4 \D4 . Clearly, λ is a fuzzy subgroup of S4 . It follows that the fuzzy subgroup Z1 (λ) is given as follows: Z1 (λ)((1)) = 1, Z1 (λ)((13)(24)) = 14 and Z1 (λ)(x) = 0 if x ∈ S4 \C. It follows easily that Z2 (λ) = e1 . Thus λ is a nilpotent fuzzy subgroup of S4 of class 2. Example 3.2.22. We now give an example of a normal fuzzy subgroup µ of S4 that is not nilpotent. Let A4 denote the alternating subgroup of S4 . Let N = {(1), (12)(34), (13)(24), (14)(23)}. Then N is a normal subgroup of S4 and N is Abelian. Let µ be the fuzzy subset of S4 deﬁned by µ(x) = 12 if x ∈ N, µ(x) = 14 if x ∈ A4 \N, and µ(x) = 0 if x ∈ S4 \A4 . Clearly, µ is a normal fuzzy subgroup of S4 . However, for n ∈ N, Zn (µ) is given by Zn (µ)((1)) = 12 , Zn (µ)(x) = 14 if x ∈ N \{(1)}), and Zn (µ)(x) = 0 if x ∈ S4 \N. Thus µ is not nilpotent.

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Theorem 3.2.23. Let λ be a fuzzy subgroup of G. Then λ is nilpotent if and only if λ has a central series. Proof. Let t = λ(e). Suppose λ has a central series λ = λ0 ⊇ λ1 ⊇ ... ⊇ λn = et . We show by induction on i that Zi (λ) ⊆ λi , i = 0, 1, . . . , n. We have Z0 (λ) = λ = λ0 and Z1 (λ) = [λ0 , λ] ⊆ λ1 . Assume Zi (λ) ⊆ λi for some i = 0, 1, . . . , n − 1, the induction hypothesis. Then by Lemma 3.2.7, Zi+1 (λ) = [Zi (λ), λ] ⊆ [λi , λ] ⊆ λi+1 . Thus Zi (λ) ⊆ λi for i = 0, 1, . . . , n by induction. Therefore, et ⊆ Zn (λ) ⊆ λn = et . Hence λ is nilpotent of class at most n. Conversely, if λ is nilpotent of class c, then λ = Z0 (λ) ⊃ Z1 (λ) ⊃ · · · ⊃ Zc (λ) = et is a central series of λ. Theorem 3.2.24. Every fuzzy subgroup of a nilpotent group is nilpotent. In fact, if the group G is not nilpotent, then G has a nontrivial fuzzy subgroup that is not nilpotent. Proof. Suppose G is nilpotent of class c. Let λ be a fuzzy subgroup of G with tip t. For i = 0, 1, ..., c, deﬁne the fuzzy subset λi of G as follows: ∀x ∈ G, λ(x) if x ∈ Zi (G), λi (x) = 0 otherwise. Then λi is a fuzzy subgroup of G and λ = λ0 ⊇ λ1 ⊇ · · · ⊇ λc is a ∗ ﬁnite chain of fuzzy subgroups of G such that (λi ) ⊆ Zi (G) for all i. Clearly, / Zi (G), then λi (a) ∧ λ(b) = 0 ≤ λc = et . Let a, b ∈ G and 0 ≤ i ≤ n − 1. If a ∈ λi+1 ([a, b]). Suppose a ∈ Zi (G). Then [a, b] ∈ Zi+1 (G). Hence λi (a) ∧ λ(b) = λ(a) ∧ λ(b) ≤ λ([a, b]) = λi+1 ([a, b]). Thus we have a central chain λ = λ0 ⊇ λ1 ⊇ ... ⊇ λc = et of λ and λ is nilpotent of class at most c by Theorem 3.2.23. Now suppose that G is not nilpotent. Consider the descending central chain G = Z0 (G) ⊇ Z1 (G) ⊇ ... ⊇ Zn (G) ⊇ ... of G. If the chain is nonterminating, then Zn (G) ⊂ Zn−1 (G) for all n ≥ 1. Deﬁne the fuzzy subset µ of G in this case as follows: µ(x) = 1 − 21n if x ∈ Zn−1 (G)\Zn (G), n ≥ 1, µ(x) = 1 if x ∈ ∩∞ n=1 Zn (G). Then µ is a nontrivial fuzzy subgroup of G which is not nilpotent. If the descending central chain terminates, then there is a nonnegative integer m such that G = Z0 (G) ⊃ Z1 (G) ⊃ ... ⊃ Zm (G) = {e}, and Zn (G) = Zm (G) for all n ≥ m. Deﬁne the fuzzy subset µ of G in this case by µ(x) = 1 − 21n if x ∈ Zn−1 (G)\Zn (G), 1 ≤ n ≤ m, µ(x) = 1 − 21m if x ∈ G\{e}, and µ(e) = 1. Clearly, µ is a fuzzy subgroup of G. It follows that Zn (µ)∗ = Zn (G) = {e} for all n ≥ 0. Therefore, µ is not nilpotent.

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Theorem 3.2.25. Every Abelian fuzzy subgroup of a group is nilpotent. Proof. Let λ be an Abelian fuzzy subgroup of G with tip t. Then H = λ∗ is an abelian subgroup of G. Consequently, Z1 (λ)∗ = [H, H] = {e}. Hence Z1 (λ) = et . Thus λ is nilpotent of class 0 or 1. Theorem 3.2.26. Let λ be a nilpotent fuzzy subgroup of G. If µ is a fuzzy subgroup of G such that µ ⊆ λ, then µ is nilpotent. Proof. If µ ⊆ λ, we have that Zn (µ) ⊆ Zn (λ) for all n ≥ 0 by Lemma 3.2.7. Therefore, if λ is nilpotent of class c, then µ is nilpotent of class at most c. Theorem 3.2.27. Let λ and µ be normal fuzzy subgroups of G. If λ and µ are nilpotent, then λ ◦ µ is nilpotent. Proof. Since λ and µ are normal, we have by repeated use of Theorem 3.2.8 that for all n ∈ N, Zn (λ ◦ µ) is contained in the sup-min product of the normal fuzzy subgroup [σ1 , . . . , σn+1 ] of G, where each σi is either λ or µ. Let r and s be the tips of λ and µ, respectively. Then t = r ∧ s is the tip of λ ◦ µ. Suppose λ and µ are nilpotent of classes c and d, respectively. Let n = c + d. Then among the σi in [σ1 , . . . , σn+1 ], either at least c + 1 of them are equal to λ or at least d + 1 of them are equal to µ. Consequently, by Theorem 3.2.17, [σ1 , . . . , σn+1 ] ⊆ Zc (λ) = er in the ﬁrst case, and [σ1 , . . . , σn+1 ] ⊆ Zd (µ) = es in the second. Therefore, Zn (λ ◦ µ) = et . Hence λ ◦ µ is nilpotent of class at most c + d. The following example shows that the above result need not be true if λ and µ are not normal. Example 3.2.28. Deﬁne the fuzzy subsets λ and µ of S3 as follows: λ((1)) = 1, λ((12)) = 12 , λ(x) = 0 if x ∈ S3 \{(1), (12)}, µ((1)) = 1, µ(x) = 13 if x ∈ {(123), (132)}, µ(x) = 0 if x ∈ {(12), (13), (23)}. Then λ and µ are Abelian fuzzy subgroups of G and therefore they are nilpotent. Even though λ, µ is a conormal pair, λ is not normal in S3 . The fuzzy subgroup λ ◦ µ of S3 is given by λ ◦ µ((1)) = 1, λ ◦ µ((12)) = 12 , λ ◦ µ(x) = 13 if x ∈ S3 \{(1), (12)}. It follows that for all n ∈ N, Zn (λ ◦ µ) = µ = e1 . Hence λ ◦ µ is not nilpotent. In view of Theorem 3.2.27, we have the following result. Theorem 3.2.29. The set of all normal nilpotent fuzzy subgroups of G is a commutative semigroup of idempotent elements with identity e1 , where the operation is the sup-min product.

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We now examine homomorphic images and preimages of nilpotent fuzzy subgroups. Theorem 3.2.30. Let f be a homomorphism of the group G into the group K. If λ is a nilpotent fuzzy subgroup of G, then f (λ) is a nilpotent fuzzy subgroup of K. Proof. By Theorem 3.2.11, we have that Zn (f (λ)) = f (Zn (λ)) for all n ∈ N. Therefore, if λ is nilpotent of class c, then f (λ) is nilpotent of class at most c. Let f be a homomorphism of the group G into the group K such that f has a nilpotent kernel. Let ν be a fuzzy subgroup of K. We show in the following example that if ν is nilpotent, then it does not necessarily follow that f −1 (ν) is nilpotent. Example 3.2.31. Let G = S3 , K = Z2 = {0, 1} and let f be the homomorphism of G onto K with kernel N = {(1), (123), (132)}. Let ν be the fuzzy subgroup of K deﬁned by ν(0) = t0 , ν(1) = t1 , where t0 > t1 .Clearly, f has the nilpotent kernel N and ν is nilpotent of class 1. However, the fuzzy subgroup f −1 (ν) of G, which is given by f −1 (ν)(x) = t0 if x ∈ N and f −1 (ν)(x) = t1 if x ∈ G\N, is not nilpotent. This can be seen by the following argument. Let µ = f −1 (ν). Then (µ, µ)(e) = t0 . Now ∀x, y ∈ G, yxa = xy implies either x or y is in aand yxa2 = xy implies either x or y is in a. Also, such x and y exist since baa = ab and ba2 a2 = a2 b. Thus (µ, µ)(a) = (µ, µ)(a2 ) = t1 . Now ∀x, y ∈ G, yxb = xy, yxab = xy, and yxa2 b = xy are impossible. Hence (µ, µ)(b) = (µ, µ)(ab) = (µ, µ)(a2 b) = 0. Therefore, (µ, µ) = [µ, µ] = Z1 (µ) = et0 . It now can easily be seen that Z2 (µ) = Z1 (µ).

3.3 Solvable Fuzzy Subgroups The commutator of a pair of fuzzy subsets of a group was deﬁned in the previous section. The notion of a solvable fuzzy group was proposed in [15], where an ascending series of fuzzy subgroups of the groups was attached to the fuzzy subgroup to deﬁne solvability. We use the notion of commutators to generate the derived chain of an arbitrary fuzzy subgroup. This derived chain is a particular descending chain of fuzzy subgroups of the underlying group. An alternative deﬁnition of a solvable fuzzy subgroup is given and some properties of solvable fuzzy subgroups are derived [18]. For a fuzzy subset λ of G, recall that λ is the fuzzy subgroup of G such that ∀x ∈ G, λ(x) = ∨{∧ {σ(t1 ), σ(t2 ), ..., σ(tn )} | x = t1 t2 ...tn , ti ∈ G, i = 1, 2, ..., n; n ∈ N},

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where σ (y) = λ(y) ∨ λ(y −1 ), y ∈ G. Recall that the commutator of two elements a and b of G is the element [a, b] = a−1 b−1 ab of G. If A and B are subsets of G, then the commutator subgroup of A and B is the subgroup [A, B] of G generated by {[a, b] | a ∈ A, b ∈ B}. For a subgroup H of G, the chain H = H (0) ⊇ H (1) ⊇ ... ⊇ H (n) ⊇ ... of subgroups of G, where H (n) = [H (n−1) , H (n−1) ], n ∈ N, is called the derived chain of the subgroup H. The subgroup H is said to be solvable if H (n) = {e} for some n ∈ N. The group G is said to be solvable if it is solvable as a subgroup of itself. Let λ be a fuzzy subgroup of a group G. Deﬁne λ(0) = λ. Let n ∈ N and suppose λ(n−1) has been deﬁned for n ≥ 1. Then deﬁne λ(n) = [λ(n−1) , λ(n−1) ]. Theorem 3.3.1. If H is a subgroup of G, then (1H )(n) = 1H (n) for all n ∈ N. Proof. The result follows from Theorem 3.2.2.

Theorem 3.3.2. Let λ be a fuzzy subgroup of G. Then for all n ∈ N, λ(n) ⊆ λ(n−1) . Proof. We prove the result by induction on n. Let x ∈ G. If x is not a commutator in G, then (λ, λ)(x) = 0 ≤ λ(x). Suppose x is a commutator in G. Then (λ, λ)(x) = ∨{λ(a) ∧ λ(b) | x = [a, b], a, b ∈ G} = ∨{λ(a−1 ) ∧ λ(b−1 ) ∧ λ(a) ∧ λ(b) | x = [a, b], a, b ∈ G} ≤ ∨{λ(a−1 b−1 ab) | x = [a, b], a, b ∈ G} = λ(x). Thus (λ, λ) ⊆ λ. Since λ is a fuzzy subgroup of G, λ(1) = [λ, λ] = [(λ, λ)] ⊆ λ = λ(0) , and thus the result holds for n = 1. Suppose λ(k) ⊆ λ(k−1) for k ∈ N, the induction hypothesis. Then by Lemma 3.2.7, we have λ(k+1) = [λ(k) , λ(k) ] ⊆ [λ(k−1) , λ(k−1) ] = λ(k) . Hence the desired result holds by induction.

Deﬁnition 3.3.3. Let λ be a fuzzy subgroup of G. The chain λ = λ(0) ⊇ λ(1) ⊇ ... ⊇ λ(n) ⊇ ... of fuzzy subgroups of G is called the derived chain of λ.

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Theorem 3.3.4. If λ is a normal (characteristic, fully invariant) fuzzy subgroup of G, then all of the fuzzy subgroups in the derived chain of λ are normal (characteristic, fully invariant) in G. Proof. The result follows from Theorem 3.2.6.

Deﬁnition 3.3.5. Let λ be a fuzzy subgroup of G with tip t. If the derived chain λ = λ(0) ⊇ λ(1) ⊇ ... ⊇ λ(n) ⊇ ... of λ is such that there is a nonnegative integer m such that λ(m) = et , then λ is called solvable. If k is the smallest nonnegative integer such that λ(k) = et , then the series λ = λ(0) ⊃ λ(1) ⊃ ... ⊃ λ(k) = et is called the derived series of λ. Theorem 3.3.6. Let H be a subgroup of G. Then H is solvable if and only if 1H is solvable. Proof. Consider the derived chain H = H (0) ⊇ H (1) ⊇ ... ⊇ H (n) ⊇ ... of H. By Theorem 3.3.1, (1H )(n) = 1H (n) for all nonnegative integers n. Thus (1H )(n) = 1{e} if and only if H (n) = {e}. The result now is clear. Example 3.3.7. The following is an example of a solvable fuzzy subgroup of the symmetric group S4 on {1, 2, 3, 4} . Let D4 = (24), (1234) = {(1), (12)(34), (13)(24), (14)(23), (13), (24), (1234), (1432)} . Then D4 is a dihedral subgroup of S4 with center C = {(1), (13)(24)}. Let λ be the fuzzy subset of S4 deﬁned by λ(x) = 1 if 1 λ(x) = if 2 1 λ(x) = if 4 λ(x) = 0 if

x∈C x ∈ (1234)\C x ∈ D4 \(1234) x ∈ S4 \D4 .

Clearly, λ is a fuzzy subgroup of S4 . It follows that the fuzzy subgroup λ(1) is given as follows: 1 λ(1) ((1)) = 1, λ(1) ((13)(24)) = 4 and λ(1) (x) = 0 if x ∈ S4 \C. It follows easily that λ(2) = e1 . Thus λ is a solvable fuzzy subgroup of S4 .

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Deﬁnition 3.3.8. Let λ be a fuzzy subgroup of G with tip t. A series λ = λ0 ⊇ λ1 ⊇ ... ⊇ λn = et of fuzzy subgroups of G is called a solvable series for λ if for all i = 0, 1, ..., n − 1, λi (a) ∧ λi (b) ≤ λi+1 ([a, b]) for all a, b ∈ G. It follows that a fuzzy subgroup λ of G is solvable if and only if [λi , λi ] ⊆ λi+1 for i = 0, 1, ..., n − 1. Theorem 3.3.9. Let λ be a fuzzy subgroup of G. Then λ is solvable if and only if λ has a solvable series. Proof. Let λ be a fuzzy subgroup of G with tip t. Suppose λ has a solvable series λ = λ0 ⊇ λ1 ⊇ ... ⊇ λn = et . We show by induction that λ(i) ⊆ λi , i = 0, 1, ..., n. We have λ(0) = λ = λ0 and λ(1) = [λ0 , λ0 ] ⊆ λ1 . Suppose λ(i) ⊆ λi for some i = 0, 1, ..., n − 1, the induction hypothesis. Then by Lemma 3.2.7, we have that λ(i+1) = [λ(i) , λ(i) ] ⊆ [λi , λi ] ⊆ λi+1 . Thus λ(i) ⊆ λi for i = 0, 1, ..., n. Therefore, et ⊆ λ(n) ⊆ λn = et . Hence λ is solvable. Conversely, if λ is solvable, then the derived series of λ is a solvable series for λ since for all a, b ∈ G, λ(i+1) ([a, b]) = [λ(i) , λ(i) ]([a, b]) = ∨{λ(i) (c) ∧ λ(i) (d)|[a, b] = [c, d]} ≥ λ(i) (a) ∧ λ(i) (b), i = 0, 1, ..... Theorem 3.3.10. Every fuzzy subgroup of a solvable group is solvable. In fact, if the group G is not solvable, then G has a nontrivial fuzzy subgroup that is not solvable. Proof. Suppose G is solvable. Let k be smallest nonnegative integer such that G(k) = {e}. Let λ be a fuzzy subgroup of G with tip t. For i = 0, 1, ..., k, deﬁne λi by ∀x ∈ G, λi (x) if x ∈ G(i) , λi (x) = 0 otherwise. Then λ = λ0 ⊇ λ1 ⊇ ... ⊇ λk is a ﬁnite chain of fuzzy subgroups of G such that (λi )∗ ⊆ G(i) for all i = 0, 1, ..., k. Clearly λk = et . Let a, b ∈ G and / G(i) , then 0 ≤ i ≤ n − 1. If a ∈ / G(i) or b ∈ λi (a) ∧ λi (b) = 0 ≤ λi+1 ([a, b]).

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If a, b ∈ G(i) , then [a, b] ∈ G(i+1) . Thus λi (a) ∧ λi (b) = λ(a) ∧ λ(b) ≤ λ([a, b]) = λi+1 ([a, b]). Hence λ = λ0 ⊇ λ1 ⊇ ... ⊇ λk = et is a solvable series for λ and so λ is solvable. Now suppose that G is not solvable. Consider the derived chain G = G(0) ⊇ G(1) ⊇ ... of G. If the chain is nonterminating, then G(n) ⊂ G(n−1) for all n ∈ N. In that case, let µ be the fuzzy subgroup of G deﬁned by µ(x) = 1 − (1/2n ) if x ∈ G(n−1) \G(n) , n ≥ 1, (i) µ(x) = 1 if x ∈ ∩∞ i=1 G .

If the derived chain terminates (but not to {e}), then there is a nonnegative integer m such that G = G(0) ⊃ G(1) ⊃ ... ⊃ G(m) = (e) , and G(n) = G(m) for all n ≥ m. Deﬁne the fuzzy subset µ of G in this case by ∀x ∈ G, µ(x) = 1 − (1/2)n if x ∈ G(n−1) \G(n) , n = 1, 2, ..., m, µ(x) = 1 − (1/2m+1 ) if x ∈ G(m) \{e}, µ(e) = 1. Clearly, µ is a nontrivial fuzzy subgroup of G. It follows that (µ(n) )∗ = G(n) = {e} for each n ≥ 0. Consequently, µ is not solvable.

Theorem 3.3.11. Suppose λ is a solvable fuzzy subgroup of G. If µ is a fuzzy subgroup of G such that µ ⊆ λ, then µ is solvable. Proof. Since µ ⊆ λ, we have by Lemma 3.2.7 that µ(n) ⊆ λ(n) for all n ≥ 0. Therefore, if λ is solvable, then µ is solvable. We now examine the homomorphic images and preimages of solvable fuzzy subgroups. Theorem 3.3.12. Let f be a homomorphism of G into a group K. If λ is a solvable fuzzy subgroup of G, then f (λ) is a solvable fuzzy subgroup of K.

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Proof. By Theorem 3.2.11(1), we have that (f (λ))(n) = f (λ(n) ) for all nonnegative integers n. Let e and e be the identities of G and K, respectively. Since λ is solvable, λ(n) = et for some n ≥ 0, where t is the tip of λ. Conse quently, (f (λ))(n) = f (et ) = et and so f (λ) is solvable. Theorem 3.3.13. Let f be a homomorphism of G into a group K. Suppose the kernel of f is solvable. If µ is solvable fuzzy subgroup of K, then f −1 (µ) is a solvable fuzzy subgroup of G. Proof. Let e and e be the identities of G and K, respectively, and let N denote the kernel of f. Let µ = µ(0) ⊇ µ(1) ⊇ ... ⊇ µ(n) = et be the derived series of µ, where t is the tip of µ. For 0 ≤ i ≤ n, set λi = f −1 (µ(i) ). Then f −1 (µ) = λ0 ⊇ λ1 ⊇ ... ⊇ λn = tN . Let N = N (0) ⊃ N (1) ⊃ ... ⊃ N (m) = {e} be the derived series of N. For 1 ≤ i ≤ m, let λn+i = tN (i) . Then [λi , λi ] = λi+1 for n ≤ i < n + m. By Theorem 3.2.12, we have that [λi , λi ] ⊆ λi+1 for all i = 0, 1, ..., n. Thus f −1 (µ) = λ0 ⊇ λ1 ⊇ ... ⊇ λn+m = et is a solvable series for f −1 (µ) and so f −1 (µ) is solvable.

References 1. W-X Gu, S-Y Li and D-G Chen, Fuzzy groups with operators, Fuzzy Sets and Systems, 66 (1994) 363-371. 73 2. K. C. Gupta and B. K. Sarma, Conormal fuzzy subgroups, Fuzzy Sets and Systems, 56 (1993) 317-322. 3. K. C. Gupta and B. K. Sarma, Operator domains of fuzzy groups, Inform. Sci. 84 (1995) 247-259. 4. K. C. Gupta and B. K. Sarma, Commutator fuzzy groups, J. Fuzzy Math. 4 (1996) 655-663. 5. K. C. Gupta and B. K. Sarma, Nilpotent fuzzy groups, Fuzzy Sets and Systems 101 (1999) 167-176. 61 6. M. Hall, Jr., The Theory of Groups, Macmillan, New York, 1959. 61 7. D. G. Kim, Characteristic fuzzy groups, Comm. Korean Math Soc. 18 (2003) 21- 29. 8. J. G. Kim, Commutative fuzzy sets and nilpotent fuzzy groups, Inform. Sci. 83 (1995) 161-174. 61 9. W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982)133-139. 10. M.A.A. Mishref, Admissible and K-invariant fuzzy subgroups, J. Fuzzy Math. 6 (1998) 811- 819.

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11. M.A.A Mishref, Restudy of fuzzy factor groups and fuzzy solvable groups, J. Fuzzy Math. 7 (1998) 311- 320. 12. N. N. Morsi, Note on normal fuzzy subgroups and fuzzy normal series of ﬁnite groups, Fuzzy Sets and Systems, 87 (1997) 255-256. 13. N. P. Mukherjee and P. Bhattacharya, Fuzzy groups: Some group-theoretic analogs, Inform. Sci. 39 (1986) 247-268. 14. S. Ray, Generated and cyclic fuzzy groups, Inform. Sci. 69 (1993) 185-200. 72 15. S. Ray, Solvable fuzzy groups, Inform. Sci. 75 (1993) 47-61. 72, 83 16. A. Rosenfeld, Fuzzy groups, J. Math. Anal.. Appl. 35 (1971) 512-517. 17. J. J. Rotman, An Introduction to the Theory of Groups, 3rd ed., Allyn and Bacon, Boston, 1984. 61 18. B. K. Sarma, Solvable fuzzy groups, Fuzzy Sets and Systems, 106 (1999) 463-467. 61, 83 19. S. I. Sidky and M.A.A. Mishref, S-fuzzy subgroups, Bulletin Calcutta Math Soc. 83 (1991) 19 - 24. 20. Wu Wang-ming, Normal fuzzy subgroups, J. Fuzzy Math. 1 (1981) 21-30 (in Chinese). 21. L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.

4 Characterization of Certain Groups and Fuzzy Subgroups

Fuzzy subgroups of Hamiltonian, solvable, P -Hall, and nilpotent groups are examined. The notions of generalized characteristic fuzzy subgroups, fully invariant fuzzy subgroups, and characteristic fuzzy subgroups are introduced. It is shown that if G is a ﬁnite group all of whose Sylow subgroups are cyclic, then a fuzzy subgroup of a group G is normal if and only if it is a generalized fuzzy subgroup of G. Normal fuzzy subgroups, quasi-normal fuzzy subgroups, (p, q)-subgroups, fuzzy cosets, fuzzy conjugates and SL(p, q)-subgroups are also presented. The results of this chapter are mainly from the work of Asaad and Abou-Zaid [9, 11, 12, 13, 14].

4.1 Fuzzy Subgroups of Hamiltonian, Solvable, P -Hall, and Nilpotent Groups Let G be a group. Recall that a fuzzy subgroup µ of G is called normal if µ(x) = µ(y −1 xy) for all x, y in G. We now consider groups whose fuzzy subgroups are normal. Deﬁnition 4.1.1. A group G is called Hamiltonian if G is not Abelian and every subgroup of G is normal. Note that the quaternion group G = x, y | x2 = y 2 , x4 = e, y −1 xy = x−1 is an example of a Hamiltonian group. Deﬁnition 4.1.2. A group that is either Abelian or Hamiltonian is called Dedekind. Theorem 4.1.3. Let G be a group. Then G is Dedekind if and only if every fuzzy subgroup of G is normal.

John Mordeson, Kiran R. Bhutani, Azriel Rosenfeld: Fuzzy Group Theory, StudFuzz 182, 91–118 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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Proof. Suppose that G is a Dedekind group. Let µ be a fuzzy subgroup of G. Then every level subgroup of G is normal since G is Dedekind. Thus by Theorem 1.3.3, µ is normal. Conversely, suppose that every fuzzy subgroup of G is normal. Every subgroup H of a group G can be regarded as a level subgroup of some fuzzy subgroup µ of G. By assumption, µ is a normal fuzzy subgroup of G. By Theorem 1.3.3, H is a normal subgroup of G. Thus G is a Dedekind group. Hamiltonian groups are characterized in the following theorem. Theorem 4.1.4. [[16], Theorem 12.5.4] A Hamiltonian group is the direct product of a quaternion group with an Abelian group in which every element is of odd order and an Abelian group of exponent two. The following corollary is an immediate consequence of Theorems 4.1.3 and 4.1.4. Corollary 4.1.5. Let G be a group. Then the following statements are equivalent. (1) G is a Hamiltonian group. (2) G is the direct product of a quaternion group with an Abelian group in which every element is of odd order and an Abelian group of exponent two. (3) G is not Abelian and every fuzzy subgroup of G is normal. We next consider cyclic groups of prime power order. Theorem 4.1.6. Let G be a group of prime power order. Then G is cyclic if and only if there exists a fuzzy subgroup µ of G such that for all x, y ∈ G, (1) if µ(x) = µ(y), then x = y, (2) if µ(x) > µ(y), then x ⊂ y. Proof. Suppose that G is cyclic of order pn , where p is a prime and n ∈ N. Let µ be the fuzzy subset of G deﬁned by ∀x ∈ G, if o(x) = pi , i = 1, 2, ..., n, µ(x) = ti µ(e) = t0 , where t0 > t1 > t2 > ... > tn . We prove that µ is a fuzzy subgroup of G. Let x, y ∈ G. Since G is a cyclic group of prime power order, it follows that x ⊆ y or y ⊆ x. Hence either xy ⊆ y or xy ⊆ x. Therefore, µ(xy) µ(x) ∧ µ(y). Clearly, µ(x−1 ) = µ(x). Thus µ is a fuzzy subgroup of G. Suppose µ(x) = µ(y). Then by the deﬁnition of µ, o(x) = o(y). Since G is a cyclic group of order pn , it follows that x = y. Thus (1) holds. Suppose µ(x) > µ(y). Then by the deﬁnition of µ, o(y) > o(x). Since G is cyclic of order pn , it follows that x ⊂ y. Thus (2) holds. Conversely, suppose there exists a fuzzy subgroup µ of G such that (1) and (2) hold for all x, y ∈ G. If G has order p, then G is cyclic. Suppose the result is true for all groups of order pm , where m = 1, ..., n − 1 for n ≥ 2, the

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induction hypothesis. Suppose o(G) = pn and let H be any proper subgroup of G. Then H is a subgroup of order pm , where 1 m < n. Clearly, the restriction of µ to H is a fuzzy subgroup of H satisfying (1) and (2). Thus H is cyclic by the induction hypothesis. Hence all proper subgroups of G are cyclic. We prove that G has a unique maximal subgroup. If this is not the case, then there exist two distinct maximal subgroups Hi = xi , where i = 1, 2. If µ(x1 ) > µ(x2 ), then H1 ⊂ H2 . However, this is contradicts the fact that H1 is maximal. Similarly, if µ(x2 ) > µ(x1 ), we have a contradiction. Thus µ(x1 ) = µ(x2 ). Hence by (1), x1 = x2 , a contradiction. Thus G has a unique maximal subgroup, say M. Hence every proper subgroup of G is contained in M. It now follows easily that G = y for any y ∈ G\M. Thus G is cyclic. We now consider fuzzy subgroups of groups of square free order. We recall from group theory that a subgroup H of a ﬁnite group G is called a Hall subgroup of G if o(H) and [G : H] are relatively prime. Theorem 4.1.7. Let G be a group of square free order. Let µ be a normal fuzzy subgroup of G. Then the following assertions hold for all x, y ∈ G. (1) o(x) | o(y) implies µ(y) µ(x). (2) o(x) = o(y) implies µ(y) = µ(x). Proof. (1) Clearly, all Sylow subgroups of G are cyclic. Thus by [[27], Theorem 12.6.17, p. 356], G = HK, H ∩ K = e, where H is a cyclic normal Hall subgroup and K is a cyclic Hall subgroup. Since H and G/H ∼ = K are solvable groups, we have that G is solvable [[27], Theorem 2.6.3, p. 39]. Clearly, x and y are Hall subgroups of G and o(x) | o(y). Since y is cyclic, it follows that y has a subgroup H such that o(H) = o(x). Thus H and x are Hall subgroups of G of the same order. Since G is solvable, it follows from Hall’s Theorem that x = g −1 Hg for some g in G [[24], Theorem 9.3.10, p. 228]. Hence x ⊆ g −1 yg and so x = g −1 y i g for some i ∈ N. Since µ is a normal fuzzy subgroup of G, µ(x) = µ(g −1 y i g) = µ(y i ) µ(y). Thus µ(x) µ(y) and so (1) holds. (2) Since o(x) = o(y), we have that o(x) | o(y) and o(y) | o(x). By (1), we have that µ(y) µ(x) and µ(x) µ(y). Thus µ(x) = µ(y) and so (2) holds. Under the assumptions of Theorem 4.1.7, it is not the case that o(x) < o(y) ⇒ µ(x) µ(y). For example, let S3 = {(1), (123), (132), (23), (13), (12)} be the symmetric group on {1, 2, 3}. Deﬁne the fuzzy subset µ of S3 as follows: µ((1)) = t0 , µ((123)) = µ((132)) = t1 , µ((23)) = µ((13)) = µ((12)) = t2 , where t0 > t1 > t2 . Then µ is a normal fuzzy subgroup of S3 . However, o((23)) < o((123)) and µ((23)) < µ((123)). Clearly, S3 is not Abelian and is of order 6. In [15], it was proved that if G is a ﬁnite Abelian group of order p1 p2 ...pr , where the pi are primes (not necessarily distinct) (i = 1, 2, ..., r), then there exists a fuzzy subgroup µ of G such that the chain of level subgroups of µ,

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e = µt0 ⊂ µt1 ⊂ ... ⊂ µtr = G is maximal, where t0 > t1 > ... > tr . A normal chain of G is a chain of subgroups, e = A0 ⊆ ... ⊆ A1 ⊆ Ar = G, where Ai−1 is a normal subgroup of Ai , i = 1, 2, ..., r. Two normal chains e = Ao ⊆ ... ⊆ Ar = G and e = B0 ⊆ ... ⊆ Bs = G of G are equivalent if and only if r = s and there exists f ∈ Sr , where Sr is the symmetric group of degree r such that Ai /Ai−1 ∼ = Bf (i) /Bf (i)−1 for 1 i r. A group G is called simple if it has contains no proper normal subgroups. A composition chain of G is a normal chain without repetition whose factors are all simple, [[27], p. 36]. We now give a generalization of the above result referred to in [15]. Theorem 4.1.8. Let G be a group of order p1 p2 ...pr , where pi is a prime, i = 1, 2, ..., r, and the pi ’s are not necessarily distinct. Then G is solvable if and only if there exists a fuzzy subgroup µ of G with exactly r level subgroups, µt0 , µt1 , ..., µtr , Im(µ) = {t0 , t1 , ..., tr }, t0 > t1 > ... > tr , such that the level subgroups form a composition chain. Proof. Suppose that G is solvable. The proof is by induction on r. If r = 1, then G is cyclic of prime order. Deﬁne the fuzzy subset µ of G by µ(e) = t0 and µ(x) = t1 for all x ∈ G, x = e, where 0 ≤ t1 < t0 ≤ 1. Then µ is a fuzzy subgroup of G. Assume that r 2. By [[16], Theorem 9.2.3, p. 139], G = G0 ⊃ G1 ⊃ G2 ⊃ ... ⊃ Gr = e , where Gi is a normal subgroup of Gi−1 and [Gi−1 : Gi ] is a prime, i = 1, 2, ..., r. By induction on r, it follows that there exists a fuzzy subgroup µ of G1 such that µt0 , µt1 , ..., µtr−1 are the level subgroups of µ, where Im(µ) = {t0 , t1 , ..., tr−1 }, t0 > t1 > ... > tr−1 > 0, and the level subgroups form a composition chain. Set µ(x) = tr for all x ∈ G\G1 , where tr < tr−1 . Then µ is a fuzzy subgroup of G with exactly r level subgroups of µ. Hence µ is the desired fuzzy subgroup of G. Conversely, let µ be a fuzzy subgroup of G such that µt0 , µt1 , ..., µtr are its level subgroups, Im(µ) = {t0 , t1 , ..., tr }, t0 > t1 > ... > tr , and the level subgroups form a composition chain. Then µti−2 µti and [µti : µti−1 ] is a prime. By [[16], Theorem 9.2.3, p. 139], G is solvable. We now make use of the Jordan-Holder Theorem, [[27], Theorem 2.5.8], which states that if G is a ﬁnite group, then any two composition chains of G are equivalent. We have the following corollary. Corollary 4.1.9. Suppose that G is a ﬁnite group and that G has a composition chain e = A0 ⊂ A1 ⊂ ... ⊂ Ar = Gr , where Ai /Ai−1 is cyclic of prime order, i = 1, 2, ..., r. Then there exists a composition chain of level subgroups of some fuzzy subgroup µ of G and this composition chain is equivalent to e = A0 ⊂ A1 ⊂ ... ⊂ Ar = G.

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Proof. Since G is a ﬁnite group and the factor groups in a composition chain from G to e are cyclic of prime order, we have by [[16], Theorem 9.2.3, p. 139] that G is solvable of order p1 p2 ...pr , where the pi ’s are (not necessarily distinct) primes. By Theorem 4.1.8, there exists a composition chain of level subgroups of some fuzzy subgroup µ of G. By the Jordan-Holder theorem, these two composition chains are equivalent. We now consider Sylow subgroups and P -Hall subgroups. A group is called periodic if all its elements are of ﬁnite order. Let P be a set of primes and let n ∈ N. Then n is called a P -number if every prime divisor of n is in P. A group G is called a P -group if G is periodic and x ∈ G implies o(x) is a P -number. Thus a ﬁnite group G is a P -group if and only if o(G) is a P -number. A group is a p-group if it is a P -group with P = {p}. Theorem 4.1.10. (Dedekind Theorem) Let G be a ﬁnite group that is not Abelian and whose proper subgroups are all normal. Then G = Q ⊗ C ⊗ E, where Q is the quaternion group of order 8, C is an Abelian group in which every element is of odd order and E is an Abelian group of exponent 2 or 1. Deﬁnition 4.1.11. A P -Hall subgroup of G is a P -subgroup of G whose index in G is not divisible by any prime in P. A Sylow p-subgroup of G is a maximal p-subgroup of G. A subgroup H of a ﬁnite group G is a Hall P -subgroup of G if H is a P -group and [G : H] is a P -number, where P is the complement of P in the set P of all prime numbers, i. e., P = P\P. We recall that if G is any group and x, y ∈ G, then x−1 y −1 xy is usually denoted by [x, y] and is called the commutator of x and y. We recall from Lemma 3.1.5 that if µ is a fuzzy subgroup of G, then µ(xy −1 ) = µ(e) implies µ(x) = µ(y) ∀x, y ∈ G. We also recall from Proposition 2.3.3 that if µ is a fuzzy subgroup of G, then µ([x, y]) = µ(e) for all x, y ∈ G implies µ is normal. Example 4.1.12. We now show that the converse of Proposition 2.3.3 is not true in general. Consider the symmetric group G = S3 = a, b | a3 = b2 = Deﬁne the fuzzy subset µ of G as follows: ∀x ∈ G, e, ba2 = ab.  if x = e, 1 µ(x) = 12 if x ∈ {a, a2 },  0 otherwise. Since the level sets of µ are normal subgroups of G, µ is a normal fuzzy subgroup. However, µ does not satisfy the condition that µ([x, y]) = µ(e)∀x, y ∈ G since µ(b−1 a−1 ba) = µ(a2 ) = 12 = µ(e). Now µ(e) = µ(a−1 b−1 (ba)). Thus µ(b−1 a−1 (ba)) = µ(a−1 b−1 (ba)). Hence Z(µ) = N (µ). Theorem 4.1.13. Let G be a ﬁnite group. Then G is Abelian if and only if for every fuzzy subgroup µ of G, µ([x, y]) = µ(e) for all x, y ∈ G.

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Proof. Suppose G is an Abelian group. Then for all x, y in G, [x, y] = e and hence µ([x, y]) = µ(e) for all fuzzy subgroups µ of G. Conversely, suppose that for every fuzzy subgroup µ of G, µ([x, y]) = µ(e) for all x, y in G. Let H be a proper subgroup of G and µ be a fuzzy subgroup of H. Clearly, by deﬁning an extension µ0 of µ to G by µ0 (x) = 0 for all x in G\H, it follows easily that µ0 is a fuzzy subgroup of G. Hence by hypothesis, µ0 satisﬁes the condition µ0 ([x, y]) = µ0 (e) for all x, y in G. Also, it is clear that the restriction of µ0 to H (which is µ) satisﬁes the condition for all x, y in H. Thus it follows by induction on the order of G that H is Abelian. By Proposition 2.3.3, every fuzzy subgroup of G is normal. Thus it follows that every subgroup of G is normal. Now by Theorem 4.1.10, we have that if G is not Abelian, then G = Q⊗C⊗E, where Q is the quaternion group of order 8, C is an Abelian group in which every element is of odd order, and E is an Abelian group of exponent 2 or 1. Since every subgroup of G is Abelian and Q is not Abelian, we have that E = C = e and so G = Q. This is impossible since there exists a fuzzy subgroup µ of Q which does not satisfy the condition. For example, deﬁne the fuzzy subgroup µ of Q = a, b | a4 = e, a2 = b2 , ba = a−1 b as follows:  ∀x ∈ G,  to if x = e, µ(x) = t1 if x = a2 ,  t2 otherwise, where 0 ≤ t2 < t1 < t0 ≤ 1. Then µ([a, b]) = µ(a2 ) = t1 = µ(e). Therefore, G is Abelian. We next consider the existence of normal Hall subgroups. Let p be a prime. A subgroup P of G is called a Sylow p-subgroup (or a p-sylow subgroup) of G if P is a maximal p-subgroup of G. Let Syl p (G) denote the set of all Sylow p-subgroups of G. Theorem 4.1.14. Let G be a group of order divisible by at least two distinct primes. Let S be a Sylow p-subgroup of G. Then S is normal in G if and only if there exists a fuzzy subgroup µ of G such that t1 > t2 , where t1 = ∧{µ(x) | x ∈ H} and t2 = ∨{µ(x) | x ∈ K} and where H = {x ∈ G | x has order p} and K = {x ∈ G | x has order p }, p a prime = p. Proof. Suppose that S is a normal Sylow p-subgroup of G. Then S = {e} ∪ H, where H is deﬁned as above. Deﬁne the fuzzy subset of µ of G as follows: ∀x ∈ G, t1 if x ∈ S, µ(x) = / S, t2 if x ∈ where 0 t2 < t1 1. Then it follows easily that µ is a fuzzy subgroup of G. We have that ∧{µ(x) | x ∈ H} = ∧{µ(x) | x ∈ S} = t1 > t2 = ∨{µ(x) | x∈ / S} = ∨{µ(x) | x ∈ K}. Conversely, suppose that µ is a fuzzy subgroup of G satisfying the condition in the theorem. Then µt1 = {x ∈ G | µ(x) t1 } is a level subgroup of µ. We

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prove that µt1 is a p-subgroup of G. Since every element of S is a p-element, we have that S is a subgroup of µt1 . Since the order of G is divisible by at least 2 distinct primes, there exists a prime q( = p) such that q | o(G). By Cauchy’s Theorem [[17], Theorem II 5.2], G contains an element y of order q. We prove that y ∈ / µt1 . If y ∈ µt1 , then µ(y) t1 > t2 = ∨{µ(x) | x ∈ K} µ(y), a contradiction. Thus q o(µt1 ) for any prime q( = p). Thus µt1 is a p-subgroup of G. Since S ⊆ µt1 and S is a Sylow p-subgroup of µt1 , we have S = µt1 . It follows easily that µt1 is a normal subgroup of G. Since o(x) = o(y −1 xy) for all x, y in G, we have that x ∈ µt1 implies y −1 xy ∈ µt1 for all y in G. Corollary 4.1.15. Let L be a proper P -Hall subgroup of G. Then L is a normal subgroup of G if and only if there exists a fuzzy subgroup µ of G such that t1 > t2 , where t1 = ∧{µ(x) | x is a non-trivial P -element of G} and t2 = ∨{µ(x) | x is a non-trivial P -element of G}. We now consider a class of nilpotent groups and (p, q)-subgroups. All groups in this section are assumed to be ﬁnite. We review brieﬂy some deﬁnitions and results. We refer the reader to [10] for details. A group G is called a (p, q)-group if the following conditions hold: (1) the factors of the order of G are of the form pi q j , i, j ∈ N, where p and q are distinct primes, (2) G is a minimal non-nilpotent group, i.e., a group which is not nilpotent and all its proper subgroups are nilpotent, and (3) the derived group G is the Sylow p-subgroup of G. Let π(G) denote the set of all primes dividing |G| . Recall that if G is any group and x, y ∈ G, then the x−1 y −1 xy is usually denoted by [x, y], and is called the commutator of x and y. Exp(G) denotes the exponent of G, i.e., the smallest n ∈ N such that xn = e∀x ∈ G. If H and K are two subgroups of a group G, then the subgroup [H, K] is deﬁned to be the subgroup generated by the set {[x, y] | x ∈ H, y ∈ K}. Let p be a prime. A subgroup P of G is a p-sylow subgroup (or a Sylow p-subgroup) of G if it is a maximal p-subgroup of G. Let Syl p (G) denote the set of all p-sylow subgroups of G. Let A4 denote the alternating group of degree 4. Then A4 = K4 , the Klein 4-group, where A4 is the commutator of A4 . In fact, [K4, A4 ] = K4 . Hence if we deﬁne the  fuzzy subgroup µ of A4 by ∀x ∈ A4 , if x = e,  t0 µ(x) = t1 if x ∈ K4 \e,  0 otherwise, where 0 < t1 < t0 ≤ 1, then µ([x, y]) = t1 = µ(e) for some x ∈ A4 and y ∈ K4 . This follows since A4 is a minimal non-nilpotent group. In general, we have the following result. Lemma 4.1.16. If G is a minimal non-nilpotent group, then there exists a fuzzy subgroup µ of G such that µ([x, y]) = µ(e) for some x ∈ G, y ∈ G .

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Proof. Since G is a minimal non-nilpotent group, G = P Q, where P ∈ Sylp (G) and Q ∈ Sylq (G) for some distinct primes p and q, P G and Q is cyclic and not normal in G, [27]. Thus it follows that G = P. Deﬁne a fuzzy subset µ of G as follows: ∀x ∈ G, if x = e,  t0 µ(x) = t1 if x ∈ G \{e},  0 otherwise, where 0 < t1 < t0 ≤ 1. Clearly, µ is a fuzzy subgroup of G. Also, we have [G, G ] = e else for all y ∈ G = P and x ∈ Q, we have [x, y] = e. Therefore, G = P ⊗ Q and so G is nilpotent, a contradiction. Therefore, there exist elements y ∈ G and x ∈ G such that [x, y] = e. Thus µ([x, y]) = t1 = µ(e) since [x, y] ∈ G \ e . Theorem 4.1.17. Let G be a group. If every fuzzy subgroup µ of G satisﬁes the condition µ([x, y]) = µ(e) for all x ∈ G and y ∈ G , then G is nilpotent. Proof. We prove the result by induction on the order of G. The result is clearly true if G is of order 1. Suppose G is of order n > 1 and the result is true for all groups of order less than n, the induction hypothesis. Let H be a proper subgroup of G and µ be a fuzzy subgroup H. By deﬁning an extension map µ0 of µ such that µ0 (x) = 0 for all x in G\H, it follows easily that µ0 is fuzzy subgroup of G and hence by hypothesis µ0 satisﬁes the condition µ0 ([x, y]) = µ0 (e) for all x ∈ G, y ∈ G . Clearly, the restriction of µ0 to H (which is µ) satisﬁes the condition µ0 ([x, y]) = µ0 (e) for all x ∈ H, y ∈ H . Thus H is nilpotent by the induction hypothesis. Hence every proper subgroup of G is nilpotent. Therefore, if G is not nilpotent then, by Lemma 4.1.16, there exists a fuzzy subgroup ν of G such that ν([x, y]) = ν(e) for some x ∈ G, y ∈ G . However, this contradicts the hypothesis. Thus G is nilpotent. The above theorem may be considered as a generalization of Theorem 4.1.14. The converse of Theorem 4.1.17 is not true in general. For that we have the following proposition. Proposition 4.1.18. Let G be a nilpotent group of class 3. Then there exists a fuzzy subgroup µ of G such that µ([x, y]) = µ(e) for some x ∈ G and y ∈ G . Proof. Since G is nilpotent of class 3, [G, G ] = e . Hence there exist elements x1 ∈ G, y1 ∈ G such that [x1, y1 ] = e. Deﬁne a fuzzy subset µ of G as follows:  ∀x ∈ G, if x = e,  t0 µ(x) = t1 if x ∈ G \ e ,  0 otherwise, where 0 < t1 < t0 ≤ 1. Clearly, µ is a fuzzy subgroup of G. Therefore, µ([x1 , y1 ]) = t1 = µ(e).

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Corollary 4.1.19. Let G be a group. If G is nilpotent and every fuzzy subgroup µ of G is such that µ([x, y]) = µ(e) ∀x ∈ G, y ∈ G , then G is nilpotent of class ≤ 2. Lemma 4.1.20. [7, 10]. G is nilpotent if and only if G contains no (p, q)subgroup for all p, q ∈ π(G) with p = q. Theorem 4.1.21. If for all fuzzy subgroups µ of G, µ([x, y]) = µ(e) for all x ∈ G, y ∈ G such that |y| = prime or 4, then G is nilpotent. Proof. Suppose every fuzzy subgroup of G satisﬁes the condition. Assume G is not nilpotent. Then there exists a minimal nonnilpotent subgroup H of G such that H = P Q, P ∈ Sylp (H), Q ∈ Sylq (H), p = q such that P H and Q is cyclic by Lemma 4.1.20. Thus by Lemma 4.1.16, there exists a fuzzy subgroup µ of H such that µ is not normal and µ([x1 , y1 ]) = µ(e) for some x1 ∈ H, y1 ∈ H = P. Now either p is odd or p = 2. If p is odd then Exp(P ) = p. Thus o(y1 ) = p. If p = 2, then Exp(P ) equals at most 4, [18]. Hence o(y1 ) = 2 or 4. Let µ0 be the extension of µ to G by deﬁning µ0 (x) = 0 for all x ∈ G\H. Then µ0 ([x1 , y1 ]) = µ0 (e) a contradiction to the hypothesis.

4.2 Characterization of Fuzzy Subgroups We introduce the notions of generalized characteristic fuzzy subgroups, fully invariant fuzzy subgroups, and characteristic fuzzy subgroups. We prove that if G is a ﬁnite group all of whose Sylow subgroups are cyclic, then µ is a normal fuzzy subgroup of G if and only if µ is a generalized characteristic fuzzy subgroup of G. A fuzzy subgroup µ of a group G is called a generalized characteristic fuzzy subgroup (GCFS) if for all x, y ∈ G, o(x) = o(y) implies µ(x) = µ(y), [2]. In Section 3.2 (see also [29]), a fuzzy subgroup µ of a group G is called a fully invariant (characteristic) fuzzy subgroup if µ ⊇ f (µ) for every f ∈ End(G) (f ∈ Aut(G)). We study some relationships between GCFS, fully invariant fuzzy subgroups, and characteristic fuzzy subgroups. Lemma 4.2.1. Let f be a homomorphism of a group G into a group H and let µ and ν be fuzzy subgroups of G and H, respectively. Then ν ⊇ f (µ) if and only if ν(f (x)) µ(x) for all x in G. Proof. ν ⊇ f (µ) ⇔ ν(f (x)) ≥ f (µ)(f (x))∀x ∈ G ⇔ ν(f (x)) ≥ ∨{µ(z) | f (z) = f (x)}∀x ∈ G ⇔ ν(f (x)) ≥ µ(x)∀x ∈ G. Proposition 4.2.2. Let µ be a fuzzy subgroup of a group G. Then the following statements are equivalent. (1) µ is fully invariant (characteristic).

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(2) µ(f (x)) µ(x) for all x in G and for all f in End(G) (f in Aut(G)). (3) Every level subgroup of µ is a fully invariant (characteristic) subgroup of G. Proof. We prove (2) ⇔ (3). Suppose (2) holds. Let f ∈ End(G) (Aut(G)) and t ∈ [0, µ(e)]. Then x ∈ µt ⇒ µ(x) ≥ t ⇒ µ(f (x)) ≥ t ⇒ f (x) ∈ µt . Thus f (µt ) ⊆ µt . Suppose (3) holds. Let x ∈ G and let µ(x) = t. Since f (µt ) ⊆ µt , f (x) ∈ µt and so µ(f (x)) ≥ t. Thus µ(f (x)) ≥ µ(x). Proposition 4.2.3. Let G be a group and let x, y ∈ G. If µ is a GCFS of G and o(x) | o(y), then µ(x) µ(y). Proof. Since o(x) | o(y), o(x) | o(y) and since y is a cyclic subgroup, it follows that y contains a subgroup H such that o(H) = o(x). Since H is cyclic, H = y1 for some y1 ∈ y and so y1 = y i for some positive integer i. Now o(x) = o(y1 ) and so µ(x) = µ(y1 ) = µ(y i ) µ(y) since µ is GCFS. Proposition 4.2.4. Let µ be a fuzzy subgroup of G. Then the following assertions hold. (1) If µ is a GCFS, then µ is a fully invariant. (2) If µ is a fully invariant, then µ is a characteristic. Proof. (1) Let µ be a GCFS and f be an endomorphism of G. It follows that o(f (x)) | o(x) ∀x ∈ G. Thus by Proposition 4.2.3, µ(f (x)) µ(x)∀x ∈ G. Hence by proposition 4.2.2, µ ⊇ f (µ). That is, µ is fully invariant. (2) The desired result follows since an automorphism is an endomorphism. The following examples show that the converse of Proposition 4.2.4 is not true. Example 4.2.5. Consider G = C2 ⊗ S3 , where C2 = c | c2 = e and S3 = a, b | a2 = b3 = e, ab = b2 a . Then Z(G) = C2 and hence C2 is a characteristic subgroup of G. Deﬁne the fuzzy subset µ of G as follows: ∀x ∈ G,   t0 if x = e, µ(x) = t1 if x = c,  0 otherwise, where 0 < t < t0 ≤ 1. The chain of level subgroups of µ is G ⊃ C2 ⊃ e and every subgroup in the chain is a characteristic subgroup of G. Hence by Proposition 4.2.2, µ is characteristic. However, µ is not a GCFS since o(c) = o(a) and µ(c) = µ(a). Furthermore, µ is not fully invariant since C2 is a level subgroup of µ that is not a fully invariant subgroup of G.

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Example 4.2.6. Let G = S4 , the symmetric group of degree 4 . Let A4 denote the alternating subgroup of G. It is known that N = {e, (12)(34), (13)(24), (14)(23)} is a normal subgroup of G. Deﬁne the fuzzy subset µ of G as follows: ∀x ∈ G,  t if x = e,  0   t1 if x ∈ N \ e , µ(x) = t2 if x ∈ A4 \N,    0 otherwise, where 0 < t2 < t1 < t0 ≤ 1. Clearly, the chain of level subsets of µ is S4 ⊃ A4 ⊃ N ⊃ e . Hence µ is a fuzzy subgroup of G. Moreover A4 = G (the commutator subgroup of G), N = A4 and e = N . Thus by [[27], Exercise 2.11.15 and Theorem 3.4.9], the level subgroups of µ are fully invariant subgroups of G. Therefore, by Proposition 4.2.2, µ is a fuzzy invariant fuzzy subgroup of G, but µ is not GCFS since o((12)) = o((12)(34)) and µ((12)) = µ((12)(34)). Proposition 4.2.7. Let G be a group. If {µi | i ∈ I} is a family of GCFS (resp. characteristic or fully invariant) fuzzy subgroups of G, then µ = ∩i∈I µi is a GCFS (resp. characteristic or fully invariant) fuzzy subgroup of G. Proof. By Theorem 1.2.13, µ is a fuzzy subgroup of G. First we prove that if each µi is a GCFS, then µ = ∩i∈I µi is a GCFS. Suppose o(x) = o(y), where x, y ∈ G. Then µi (x) = µi (y)∀i ∈ I. In fact, µ(x) = ∩i∈I µi (x) = ∧{µi (x) | i ∈ I} = ∧{µi (y) | i ∈ I} = µ(y) and so µ is a GCFS. Now we show that if µi is a characteristic (fully invariant) fuzzy subgroup ∀i ∈ I, then µ = ∩i∈I µi is characteristic (fully invariant). Let f ∈ Aut(G) (f ∈ End(G)). Then f (µ)(y) = f (∩i∈I µi )(y) ∨{∩i∈I µi )(x) | f (x) = y} if f −1 (y) = ∅, ={ 0 if f −1 (y) = ∅. ∨{∧{µi (x) | i ∈ I} | f (x) = y} if f −1 (y) = ∅, = 0 if f −1 (y) = ∅. ∧{∨{µi (x) | f (x) = y} | i ∈ I} if f −1 (y) = ∅, 0 if f −1 (y) = ∅. ≤ ∧{f (µi )(y) | i ∈ I} = ∩i∈I f (µi )(y) ∩i∈I (µi )(y) (since µi is characteristic (fully invariant)) = µ(y). We prove that if G is a group all of whose Sylow subgroups are cyclic, then µ is a normal fuzzy subgroup of G if and only if µ is a GCFS. Remark 4.2.8. Let G be a group all of whose Sylow subgroups are cyclic. Then by [[27], Theorem 12.6.7], G = HK, H ∩ K = e and H, K are cyclic Hall subgroups and H is normal in G. Furthermore, every element in G is of

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the form hi k j , where H = h and K = k. Thus the following properties are immediate: (1) G is a solvable group. (2) If o(H) = n, o(K) = m, then o(G) = nm and since G is solvable, it follows from Hall’s theorem [[17], Theorem 7.14, p. 104] that any two subgroups of order m (or n) are conjugate. However, H is normal if G contains only one subgroup, namely H, of order n. Theorem 4.2.9. Suppose G is a group all of whose Sylow subgroups are cyclic. Let µ be a normal fuzzy subgroup of G. Then the following properties hold ∀x, y ∈ G. (1) If o(x) | o(y), then µ(y) µ(x). (2) If o(x) = o(y), then µ(y) = µ(x). Proof. (1) Since all Sylow subgroups of G are cyclic, it follows from Remark 4.2.8 that G = HK with H ∩ K = e, where K and H are cyclic Hall subgroups of G such that H is normal in G. Furthermore, every element of G is of the form hi k j , where H = h and K = k, k m = e = hn , (n, m) = 1 and o(G) = nm. Suppose that x, y ∈ G are such that o(x) | o(y). Then we have the following three cases. Case 1: Let y ∈ H. By Remark 4.2.8, H is the only normal cyclic subgroup of order n. Thus it follows that G has a unique subgroup of any order that divides n. Therefore, H ⊃ y ⊃ x. Hence x = y i for some integer i. Thus µ(x) = µ(y i ) µ(y). Case 2: Let y ∈ K. Consider the following two subcases. Subcase 2.1: Suppose y ∈ K and x ∈ K. Then y ⊃ x and x = y j for some integer j. Therefore, µ(x) = µ(y j ) µ(y). Subcase 2.2: Suppose y ∈ K and x ∈ / K. Then it follows that x ∈ z −1 Kz for some z ∈ H by Remark 4.2.8. Therefore, there exists x1 ∈ K such that x1 = zxz −1 . Hence o(x) = o(x1 ). Thus y ⊃ x1 . Now x1 = y i for some integer i. Since µ is normal, µ(x) = µ(zxz −1 ) = µ(x1 ) = µ(y i ) µ(y). Case 3: Suppose y ∈ / H and y ∈ / K. If o(y) = nm, then G is cyclic and is generated by y. Thus µ(y) ≤ µ(x). Suppose o(y) = nm. We consider the following two subcases. Subcase 3.1: Suppose o(y) = m , m | m. Then z −1 Kz ⊃ y for some z ∈ H and there exists x1 ∈ z −1 Kz such that x1 = z1−1 xz1 for some z1 ∈ H. Thus y ⊃ x1 . Hence x1 = y i for some integer i. Therefore, µ(x) = µ(z1−1 xz1 ) = µ(x1 ) = µ(y i ) µ(y). Subcase 3.2: Suppose o(y) = m1 n1 , where m1 n1 | mn and (m1 , n1 ) = 1. Then by [[15], Lemma 3.2.1, p. 39], y has a unique representation y = y1 y2 = y2 y1 , where o(y1 ) = n1 , o(y2 ) = m1 . Furthermore, both y1 and y2 are powers of y, i.e., y1 = y j1 , y2 = y j2 for integers j1 , j2 . Now since o(x) | o(y), we have that o(x) = m2 n2 , where m2 | m1 and n2 | n1 . Thus we have three diﬀerent possibilities:

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(3.2.1) m2 = 1 and so o(x) = n2 : Since y1 is a ﬁnite cyclic group of order n1 and n2 | n1 , it follows that y1 has a unique subgroup of order n2 , namely x . Therefore, y1 ⊃ x and so x = y1k for some integer k. Thus µ(x) = µ(y1k ) = µ((y j1 )k ) = µ(y j1 k ) µ(y). (3.2.2) n2 = 1 and so o(x) = m2 : Then y2 ∈ v −1 Kv for some v ∈ H and there exists z2 ∈ H such that z2−1 xz2 ∈ v −1 Kv. Now since o(z2−1 xz2 ) = o(x), it follows that z2−1 xz2 ∈ y2 . Hence z2−1 xz2 = y2q for some integer q. Thus it follows that µ(x) = µ(z2−1 xz2 ) = µ(y2q ) = µ((y α2 )q ) = µ(y α2 q ) µ(y). (3.2.3) o(x) = m2 n2 , m2 = 1 = n2 and (n2 , m2 ) = 1 : Then by [[15], Lemma 3.2.1, p. 39], x has a unique representation x = x1 x2 = x2 x1 , o(x1 ) = n2 and o(x2 ) = m2 . Clearly, H ⊃ y1 ⊃ x1 . Hence x1 = y1e1 for some integer e1 . Also, y2 ∈ w1−1 Kw1 and x2 ∈ w2−1 Kw2 for some w1, w2 ∈ H. Thus there exists u ∈ H such that u−1 x2 u ∈ w−1 Kw1 . Hence u−1 x2 u ∈ y2 . Thus u−1 x2 u = y2e2 for some integer e2 . Therefore, we have that µ(x) = µ(x1 x2 ) µ(x1 ) ∧ µ(x2 ) µ(y1e1 ) ∧ µ(u−1 x2 u) (since µ is normal) µ((y j1 )e1 ) ∧ µ(y2e2 ) µ(y j1 e1 ) ∧ µ(y j2 e2 ) µ(y) ∧ µ(y) µ(y). Thus (1) holds. (2) Since o(x) = o(y), we have that o(x) | o(y) and o(y) | o(x). By (1), we have that µ(x) µ(y) and µ(x) µ(y). Hence µ(x) = µ(y). Thus (2) holds. Corollary 4.2.10. Let G be a group all of whose Sylow subgroups are cyclic. Let µ be a fuzzy subgroup of G. Then µ is normal if and only if µ is a GCFS. The following example shows that the normality assumption in Theorem 4.2.9 cannot be removed. Example 4.2.11. Let G = S3 , the symmetric group on {1, 2, 3}. Let e = (1), a = (12), b = (123). Then b2 = (132), ab = (13), ab2 = (23). Clearly, S3 is a group all of whose Sylow subgroups are cyclic. Deﬁne the fuzzy subset µ of S3 as follows:  ∀x ∈ G,  t0 if x = e, µ(x) = t1 if x = a,  0 otherwise, where 0 < t1 < t0 ≤ 1.Then µ is a fuzzy subgroup of G which is not normal. Also, µ is not GCFS since µ(a) = µ(ab) and o(a) = o(ab). Corollary 4.2.12. Let G be a dihedral group of order 2n with n odd and let µ be a fuzzy subgroup of G. Then µ is normal if and only if µ is GCFS. Proof. Now G = a, b | an = e = b2 , ba = a−1 b . It follows that a is normal in G, a ∩ b = e and G = ab. Hence it follows that all Sylow subgroups of G are cyclic. Thus by applying Corollary 4.2.10, we obtain the desired result.

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The following example shows that Corollary 4.2.12 is not true in general if the dihedral group G is of order 2n, where n is even. of order 8. For this group, Example 4.2.13. Let G = D4 , the dihedral group n = 4 and G = a, b | a4 = e = b2 , ba = a−1 b . Deﬁne the fuzzy subset µ of G as follows:  ∀x ∈ G, t0 if x = e,    t1 if x = a2 , µ(x) = t2 if x = a, a3 ,    0 otherwise, where 0 < t2 < t1 < t0 ≤ 1. Clearly, the level subsets of µ form the chain, e a2 a G, which is a normal chain of subgroups. Hence µ is a normal fuzzy subgroup of G, but µ is not a GCFS since o(b) = o(a2 ) and µ(b) = µ(a2 ).

4.3 Quasi-normal and Normal Fuzzy Subgroups We now consider normal fuzzy subgroups, quasinormal fuzzy subgroups, (p, q)subgroups, fuzzy cosets, fuzzy conjugates and SL(p, q)-subgroups. We show that (Q(G), ◦) is a commutative idempotent semisimple semigroup, where Q(G) is the set of all quasinormal fuzzy subgroups of G and ◦ is the product operation of fuzzy subgroups. We prove some results about fuzzy conjugate subgroups. The concepts of the core and the closure of a fuzzy subgroup of a group G are introduced. We give some equivalent conditions for G/µ to be nilpotent of class at most n. We also give a characterization of groups which have normal p-complements in terms of certain fuzzy subgroups. Throughout the remainder of the chapter all groups are assumed to be of ﬁnite order. Let G be a group and µ, ν be fuzzy subgroups of G. Recall that the product µ ◦ ν is deﬁned as follows: µ ◦ ν(x) = ∨{µ(y) ∧ ν(z) | y, z ∈ G, x = yz} for all x ∈ G. The operation ◦ is associative. Also, it is commutative if and only if µ ◦ ν is a fuzzy subgroup of G for all fuzzy subgroups µ, ν of G. Moreover, we have the following result. Theorem 4.3.1. [4] (N F (G), ◦) is a commutative idempotent semisimple semigroup, where N F(G) is the set of all normal fuzzy subgroups of a group G. Deﬁnition 4.3.2. A subgroup H of a group G is called quasinormal if H permutes with every subgroup of G.

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A subnormal series of a group G is a chain of subgroups G = G0 ⊇ G1 ⊇ ... ⊇ Gn such that Gi+1 is a normal subgroup of Gi , i = 0, 1, ..., n − 1. A subgroup H is called subnormal in G if there is a subnormal series from G to H. Theorem 4.3.3. [19] If H is a quasinormal subgroup of G, then H is subnormal in G. Recall from Lemma 3.1.5 that ∀x, y ∈ G, µ(xy −1 ) = µ(e) implies µ(x) = µ(y). Further, from Section 3.2 that Z0 (G) = G and Zi (G) = [Zi−1 (G), G] for i = 1, 2, ... and from Section 3.3 that G(0) = G and G(i) = [G(i−1) , G(i−1) ] for i = 1, 2, .... Recall also that for all i ∈ N, Zi (G) ⊇ G(i) . Theorem 4.3.4. [24] The following statements are equivalent. (1) G is nilpotent of class at most n, (2) Zn (G) = e. Recall that a group G is called a (p, q)-group if the following conditions hold: (1) The order of G involves only the prime factors p and q. (2) G is a minimal non-nilpotent group, i.e., a group which is not nilpotent and all its proper subgroups are nilpotent. (3) The derived group G is the Sylow p-subgroup of G. We also recall that if G is a group and x, y ∈ G, then the element x−1 y −1 xy is usually denoted by [x, y] and is called the commutator of x and y. Exp(G) stands for exponent of G, the smallest positive integer n such that xn = e for all x ∈ G. Deﬁnition 4.3.5. The group of all nonsingular 2 × 2 matrices over the ﬁeld Z3 with determinate equal to 1 is called an SL(2, 3) group. An SL(2, 3) group contains only one Sylow 2-subgroup. Theorem 4.3.6. [8] Let S ∈ Syl2 (G). Then S G if and only if G contains no (p, 2)-subgroup for all p ∈ π(G)\{2},where π(G) is the set of all distinct primes that divide o(G)). Deﬁnition 4.3.7. A fuzzy subgroup µ of G is called quasinormal if its level subgroups are quasinormal subgroups of G. Proposition 4.3.8. Every normal fuzzy subgroup of G is quasinormal. Proof. The result is immediate since any normal subgroup is quasinormal.

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Example 4.3.9. We show that the converse of Proposition 4.3.8 is not true. p p2 −1 1+p , where p is Consider the group G = x, y | x = y = e, xyx = y an odd prime and e is the identity element of G. It is known that x is a quasinormal subgroup, but not normal [26]. Deﬁne a fuzzy subset µ of G as follows: ∀x ∈  G, if y = e,  t0 µ(y) = t1 if y ∈ x\e,  0 otherwise, where 0 < t1 < t0 ≤ 1. Clearly, µ is a quasinormal fuzzy subgroup of G since its level subgroups G, x , e are quasinormal subgroups of G. However, µ is not a normal fuzzy subgroup of G since x is not normal in G. Let µ be a fuzzy subgroup of G and let t ∈ [0, 1]. Deﬁne the fuzzy subset tµ of G by (tµ)(x) = tµ(x) for all x ∈ G. Let Im(µ) = {t0 , t1 , ..., tn }, where t0 > t1 > ... > tn . We may write µ in the following form: n ti (1µti − 1µti−1 ), where 1µt0 −1 (x) = 0∀x ∈ G. µ= i=0

Lemma 4.3.10. Let µ be a fuzzy subgroup of G and let H, K be subgroups of G. Let ti ∈ Im(µ), i = 0, 1, ..., n. Then the following assertions hold. (1) Let t ∈ [0, 1]. Then tµ is a fuzzy subgroup of G. Furthermore, tµ and µ have the same level subgroups if t = 0. (2) 1HK = 1H ◦ 1K . (3) (1µti − 1µti−1 ) ◦ 1K = 1µti ◦ 1K − 1µti−1 ◦1K . (4) (µti −1µti−1 ) ◦ ν = 1µti ◦ ν − 1µti−1 ◦ν for every fuzzy subgroup ν of G. (5) (ti 1µri ) ◦ 1K = ti (1µti ◦ 1K ) = 1µti ◦ (ti 1K ). (6) [ti (1µti − 1µti−1 ) + ti−1 (1µti−1 − 1µti−2 )] ◦ 1K = [ti (1µti − 1µti−1 ) ◦ 1K + ti−1 (1µti−1 − 1µti−2 )] ◦ 1K . n n (7) [ i=0 ti (1µri − 1µti−1 )] ◦ 1K = i=0 [ti (1µti − 1µti−1 ) ◦ 1K ]. n n (8) [ i=0 ti (1µri − 1µti−1 )] ◦ ν = i=0 [ti (1µti −µti−1 ) ◦ ν] for every fuzzy subgroup ν of G. Proof. (1) This follows from deﬁnitions. (2) Let x ∈ G. Then 1KH (x) = 1 ⇔ x ∈ HK ⇔ x = x1 x2 , where x1 ∈ H, x2 ∈ K ⇔ 1H ◦ 1K (x1 x2 ) = 1 ⇔ 1H ◦ 1k (x) = 1. Also, 1HK (x) = 0⇔x∈ / HK ⇔ x cannot be written as a product x1 x2 , with x1 ∈ H, x2 ∈ / K) or (x1 ∈ / H and x2 ∈ K) or (x1 ∈ / H and K ⇔ x = x1 x2 (x1 ∈ H and x2 ∈ x2 ∈ / K) ⇔ 1H ◦ 1K (x) = 0. (3) Let x ∈ G. Then (1µti − 1µti−1 ) ◦ 1K (x) if x = yz, y ∈ µti \µti−1 , z ∈ K, = 0 otherwise, / µti−1 , z ∈ K), 1 if x = yz (y ∈ µti , z ∈ K) and (y ∈ = 0 otherwise,

4.3 Quasi-normal and Normal Fuzzy Subgroups

107

= 1µti ◦ 1K (x) − 1µti−1 ◦ 1K (x) = (1µti ◦ 1K (x) − 1µti−1 ◦ 1K )(x). (4) - (8) : An argument similar to that in (3) establishes (4) - (8). We note that in Lemma 4.3.10 parts (3) − (8) are valid if we act from the right by ◦, e.g., (4) ν ◦ (1µti − 1µti−1 ) = ν ◦ 1µti − ν ◦ 1µti−1 . Proposition 4.3.11. Let K be a subgroup of G. Then K is quasinormal if and only if 1K is a quasinormal fuzzy subgroup of G. Moreover, K is quasinormal if and only if 1K ◦ 1H = 1H ◦ 1K for all subgroups H of G. Proof. The result follows from Lemma 4.3.10 (2) and the deﬁnition of quasinormality. Proposition 4.3.12. Let K be a subgroup of G. Then K is quasinormal if and only if 1K ◦ µ = µ ◦ 1K for all fuzzy subgroups µ of G. Proof. Let K be a subgroup of G such that 1K ◦ µ = µ ◦ 1K for every fuzzy subgroup µ of G. Then 1K ◦ 1H = 1H ◦ 1K for every subgroup H of G. Thus it follows by Lemma 4.3.10 (2) that 1KH = 1HK . Hence KH = HK. Thus K is quasinormal. Conversely, let K be a quasinormal subgroup of G and µ be any fuzzy subgroup of G. Then for ti ∈ Im(µ), i = 1, ..., n, where 1µt0 −1 (x) = 0∀x ∈ G, n µ ◦ 1K = ( ti (1µti − 1µti−1 )) ◦ 1K =(

n

i=0

= =

n

i=0

ti 1µti − ti 1µti−1 )) ◦ 1K

((ti 1µti ) ◦ λK − (ti 1µti−1 ) ◦ 1K )

(by Lemma 4.3.10)

(ti (1K ◦ 1µti ) − ti (1K ◦ λµti−1 ))

(by Proposition 4.3.11)

i=0 n i=0

= 1K ◦ (

n

i=0

= 1K ◦ µ.

ti (1µti − 1µti−1 ))

(by Lemma 4.3.10)

As in [6], deﬁne the relation ∼ on F(G) as follows: ∀µ, µ ∈ F(G), µ ∼ µ if and only if for all x, y in G, µ(x) µ(y) ⇔ µ (x) µ (y). Then ∼ is an equivalence relation on F(G). Moreover, if µ, µ are fuzzy subgroups of G, then µ ∼ µ ⇔ µ and µ have the same set of level subgroups. Theorem 4.3.13. Let µ be a fuzzy subgroup of G with Im(µ)={t0 , t1 , ..., tn }. Then µ is quasinormal if and only if µ ◦ ν = ν ◦ µ for all fuzzy subgroups ν of G.

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4 Characterization of Certain Groups and Fuzzy Subgroups

Proof. Suppose µ is a quasinormal fuzzy subgroup of G. Then the level subgroups µti of µ are quasinormal subgroups of G and so 1µti is quasinormal for i = 0, 1, ..., n. Therefore, 1µti ◦ ν = ν ◦ 1µti for every fuzzy subgroup ν of G and ∀i = 0, 1, .., n. Hence n µ ◦ ν = ( ti (1µti − 1µti−1 )) ◦ ν =( = = =

i=0

n

i=0 n

ti (1µti − 1µti−1 ) ◦ ν)

(by Lemma 4.3.10)

(ti (1µti ◦ ν) − ti (1µti−1 ◦ ν))

(by Lemma 4.3.10)

(ti (ν ◦ 1µti ) − ti (ν ◦ 1µti−1 ))

(by Proposition 4.3.12)

i=0 n i=0 n

(ν ◦ ti (1µti − 1µti−1 ))

i=0

=ν◦(

n

i=0

(by Lemma 4.3.10)

ti (1µti − 1µti−1 ))

(by Lemma 4.3.10)

= ν ◦ µ. Conversely, assume that µ is a fuzzy subgroup of G such that µ ◦ ν = ν ◦ µ for every fuzzy subgroup ν of G. Then µ ◦ 1K = 1K ◦ µ for every subgroup K of G. Deﬁne a fuzzy subgroup µ of G as follows: for 0 < s < 1, Im(µ ) = {s0 , s, s2 , ..., sn }, where µs0 = µt0 , µs = µt1 , µs2 = µt2 , ..., µsn = µtn = G. Then µ and µ have the same chain of level subgroups. Hence if µ is quan si (1µ i − 1µ i−1 ) , where sinormal, then µ is by deﬁnition. Now µ = Σi=0 s s 1µ 0−1 (x) = 0∀x ∈ G. Thus for all x ∈ G, s µ ◦ 1K (x) = 1K ◦ µ (x) n n ⇒ ( si (1µ lsi − 1µ si−1 )) ◦ 1K (x) = 1K ◦ ( si (1µ si − 1µ si−1 ))(x) i=0

⇒

n

i=0

i=0 i

s (((1

µ i s

◦ 1K − 1K ◦ 1

µ i s

) − (1

µ i−1 s

◦ 1K − 1K ◦ 1µ i−1 )))(x) = 0 s

⇒ (((1µ i ◦1K −1K ◦1µ i )−(1µ i−1 ◦1K −1K ◦1µ i−1 )))(x) = 0, i = 0, 1, ..., n s s s s ⇒ (1µ i ◦ 1K − 1K ◦ 1µ i )(x) = (1µ i−1 ◦ 1K − 1K ◦ 1µ i−1 )(x), i = 0, 1, ..., n s s s s ⇒ (1µ i ◦ 1K − 1K ◦ 1µ i )(x) = 0, i = 0, 1, ..., n, s s for otherwise there exists x ∈ G such that (1µ i ◦ 1K − 1K ◦ 1µ i )(x) = 1 or s s −1, say 1, and so 1µ i ◦ 1K (x) = 1 and 1K ◦ 1µi (x) = 0 for all 0 i n. s Hence 1G ◦ 1K (x) = 1 and 1K ◦ 1G (x) = 0 and so 1GK (x) = 1, 1KG (x) = 0. Therefore, x ∈ GK and x ∈ / KG, a contradiction. Thus it follows that 1µ i ◦ 1K = 1K ◦ 1µ i s s and so (by Lemma 4.3.10(2)). 1µ i K = 1Kµ i for all 0 i n s s Hence µsi K = Kµsi for every subgroup K of G and 0 i n. Thus µsi is quasinormal subgroup for all 0 i n. Hence µti is a quasinormal subgroup for all 0 i n. Therefore, µ is quasinormal of G.

4.3 Quasi-normal and Normal Fuzzy Subgroups

109

Recall that if µ and ν are two fuzzy subgroups of G, then µ ◦ ν is a fuzzy subgroup of G if and only if µ ◦ ν = ν ◦ µ. As a consequence of this result, we have the following two corollaries. Corollary 4.3.14. If µ is a quasinormal fuzzy subgroup and ν is any fuzzy subgroup of G such that Im(µ) is ﬁnite, then µ ◦ ν is fuzzy subgroup of ν. Corollary 4.3.15. If µ and ν are quasinormal fuzzy subgroups of G with ﬁnite images, then µ ◦ ν is a quasinormal fuzzy subgroup of G. Proof. By Corollary 4.3.14, it follows that µ ◦ ν is a fuzzy subgroup of G. Let γ be a fuzzy subgroup of G. Then since ◦ is a associative and ν and µ are quasinormal, (µ ◦ ν) ◦ γ = µ ◦ (ν ◦ γ) = µ ◦ (γ ◦ ν) = (µ ◦ γ) ◦ ν = (γ ◦ µ) ◦ ν = γ ◦ (µ ◦ ν) . Thus µ ◦ ν is quasinormal by Theorem 4.3.13. Corollary 4.3.16. Let Q(G) denote the set of all quasinormal fuzzy subgroups of G. Then (Q(G), ◦) is a commutative idempotent semigroup. Proof. By Corollary 4.3.15, it follows that Q(G) is closed under operation ◦ and that ◦ is commutative. Since ◦ is an associative, (Q(G), ◦) is a semigroup. Also, for any fuzzy subgroup µ of G, µ ◦ µ = µ. Deﬁnition 4.3.17. Let S be a semigroup and M be a nonempty subset of S. (1) M is called a left (right) ideal of S if M ⊇ SM (M ⊇ M S). M is called an ideal if it is a left and right ideal. (2) S is called semisimple if M 2 = M for every ideal M of S. Deﬁnition 4.3.18. Let A and B be two subsets of Q(G). Deﬁne A ◦ B = {µ ◦ ν | µ ∈ A, ν ∈ B} and A2 = A ◦ A. Theorem 4.3.19. (Q(G), ◦) is a commutative idempotent semisimple semigroup. Proof. By Corollary 4.3.16, it suﬃces to show that (Q(G), ◦) is semisimple. Let A be an ideal in Q(G). Then A ⊇ A2 . Let µ ∈ A. Then µ = µ ◦ µ ∈ A2 and so A2 ⊇ A. Thus A2 = A for every ideal A in Q(G). Hence (Q(G), ◦) is semisimple. Note that (N F (G), ◦) is a subsemigroup of (Q(G), ◦), where N F(G) is the set of all normal fuzzy subgroups of G.

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4 Characterization of Certain Groups and Fuzzy Subgroups

Deﬁnition 4.3.20. A fuzzy subgroup is called subnormal if its level subgroups are subnormal. Theorem 4.3.21. Any quasinormal fuzzy subgroup of G is subnormal. Proof. The result is immediate by Theorem 4.3.3 and the deﬁnitions of fuzzy quasi-normality and subnormality. We now show that the converse of Theorem 4.3.21 is not true: Let G = A4 , the alternating group on {1, 2, 3, 4}. Deﬁne the fuzzy subset µ of G as follows: ∀x ∈ G,  t0 if x = (1) = e,    t1 if x = (12)(34), µ(x) = t2 if x ∈ N \(12)(34),    0 otherwise, where 0 ≤ t2 < t1 < t0 ≤ 1 and N = {(1), (12)(34), (13)(24), (23)(14)}. Then µ is a fuzzy subgroup since its level subsets are subgroups. Now the level subgroups of µ form a normal chain, (1) (12)(34) N A4 = G. Thus µ is subnormal. However, µ is not quasinormal since (12)(34) is a level subgroup of µ that is not quasinormal. Corollary 4.3.22. Let G be a group of square free order and µ be a fuzzy subgroup of G. Then µ is quasinormal if and only if µ is normal. Proof. Suppose µ is quasinormal. Then the level subgroups of µ are quasinormal subgroups in G. Thus by Theorem 4.3.3, the level subgroups of µ are subnormal. However, since G in square free order, every level subgroup µt , t ∈ Im(µ), is a Hall subgroup of G. Hence µt is normal for all t ∈ Im(µ) and thus µ is normal. The converse follows from Proposition 4.3.8. We now consider conjugate fuzzy subgroups. Recall that if µ and ν are fuzzy subsets of a group G, then µ is said to be conjugate to ν if there exists x ∈ G such that ν = x−1 µx. The notation µx = x−1 µx is often used, where µx (g) = µ(x−1 gx)∀x ∈ G. It follows that (µx )y = µxy for all x and y in G. This notion of conjugacy is an equivalence relation on FP(G). Suppose µ ∈ F(G). Then µg ∈ F(G)∀g ∈ G. Also, µ is normal if and only if µg is normal ∀g ∈ G. It follows easily that for all x ∈ G, µ and µx have, in order of inclusion, conjugate level subgroups of G. Moreover, the length of the chains of their level subgroups are equal. Example 4.3.23. Let G = A4 , the alternating group on {1, 2, 3, 4}. Then G = {(1), (12)(34), (13)(24), (14)(23), (123), (132), (142), (124), (234), (243), (134), (143)}. Deﬁne the fuzzy subset µ as follows: ∀x ∈ G,    t0 if x = (1) = e,  t1 if x = (12)(34), µ(x) = t2 if x ∈ {(13)(24), (14)(23)},    0 otherwise,

4.3 Quasi-normal and Normal Fuzzy Subgroups

111

where 0 < t2 < t1 < t0 ≤ 1. Then µ is a fuzzy subgroup of G and the chain of level subgroups of µ is (1) (12)(34) {(1), (12)(34), (13)(24), (23)(14)} G. Let x = (123). Then µx ((12)(34)) = µ((132)(12)(34)(123)) = µ((13)(24)) = t1 . We see it follows that (1) (13)(24) {(1), (12)(34), (13)(24), (23)(14)} G. is the chain of level subgroups of µx . The chains have the same length and the level subgroups of µ and µx , in order of inclusion, are conjugate. Let µ be a fuzzy subgroup of G. Let cl (µ) denote the set of all fuzzy subgroups ν of G which are conjugate to µ. Theorem 4.3.24. If µ is a fuzzy subgroup of G, then [G : N (µ)] = |cl(µ)|. Proof. Let x, y ∈ G. Then µx = µy if and only if y −1 x ∈ N (µ). Since y −1 x ∈ N (µ) if and only if xN (µ) = yN (µ), the desired result now follows. We now obtain, for fuzzy subsets, an analog of a consequence of the following lemma. Lemma 4.3.25. [[27], Exercise 3.3.22(a)] Let f be a homomorphism of G onto a group H and let S is a subset of G. Then f (cl(S)) = cl(f (S)). Lemma 4.3.26. Let f be a homomorphism of G onto a group H. If µ is a fuzzy subset of G, then f (cl(µ)) = cl(f (µ)). Proof. We have that ν ∈ cl (f (µ)) ⇔ ν = (f (µ))f (x) for some x ∈ G ⇔ ν(f (y)) = f (µ)f (x) (f (y)) ∀y ∈ G ⇔ ν(f (y)) = f (µ)(f (x−1 )f (y)f (x))∀y ∈ G ⇔ ν(f (y)) = f (µ)(f (x−1 yx))∀y ∈ G ⇔ ν(f (y)) = ∨{µ(z) | f (z) = f (x−1 yx), z ∈ G}∀y ∈ G ⇔ ν(f (y)) = ∨{µ(z) | f (x−1 zx) = f (y), z ∈ G}∀y ∈ G ⇔ ν(f (y)) = ∨{µ(x−1 ux) | f (u) = f (y), u ∈ G}∀y ∈ G ⇔ ν(f (y)) = ∨{µx (u) | f (u) = f (y), u ∈ G}∀y ∈ G ⇔ ν(f (y)) = f (µx )(f (y))∀y ∈ G ⇔ ν = f (µx ) ⇔ ν ∈ f (cl (µ)). Let µ be a fuzzy subset of G. Deﬁne the subset C(µ) as follows: C(µ) = {x ∈ G | µ([x, y]) = µ(e) for all y ∈ G}. Clearly, N (µ) = {x ∈ G | µ(x−1 yx) = µ(y) ∀ y ∈ G} = {x ∈ G | µ(xy) = µ(yx) ∀ y ∈ G} ⊇ C(µ). In general, C(µ) = N (µ). For example, let G = D4 = a, b | a4 = 1 = b2 , ba = a−1 b , the dihedral group  of degree 4. Deﬁne the fuzzy subset µ as follows: ∀x ∈ G, t0 if x = e,    t1 if x = a2 , µ(x) = t2 if x ∈ {a, a3 },    0 otherwise,

112

4 Characterization of Certain Groups and Fuzzy Subgroups

where 0 < t2 < t1 < t0 ≤ 1. Then µ is a normal fuzzy subgroup since its level subsets are normal subgroups of G, and in fact, e a2 a G. 2 Therefore, / C(µ). Now (µ) = G. Now µ([a, b]) = µ(a ) = t0 . Thus a, b ∈ 2N C(µ) = a . Hence N (µ) ⊃ C(µ). Proposition 4.3.27. If µ is a fuzzy subgroup of G, then C(µ) is a subgroup of G. Proof. Since e ∈ C(µ), C(µ) is not empty. Let x, y ∈ C(µ). Then µ([x, z]) = µ(e) and µ([y, z]) = µ(e) for all z ∈ G. We have that µ([xy, z]) = µ([x, z]y [y, z]) ([x, z]y ) ∧ µ([y, z]) (µ([x, z]y ) (since µ([y, z]) = µ(e)) µy ([x, z]) µ([x, z]) (since y ∈ N (µ)) = µ(e). Thus xy ∈ C(µ). Let x ∈ C(µ). Then µ([x, z]) = µ(e) for all z ∈ G. Hence µ([x−1 , z]) = µ(xz −1 x−1 z) = µ(xz −1 x−1 zxx−1 ) −1 = µx (z −1 x−1 zx) (since x−1 ∈ NG (µ)) = µ([z, x]) = µ([x, z]−1 ) = µ([x, z]) = µ(e). Therefore, x−1 ∈ C(µ). Thus C(µ) is a subgroup of G. Deﬁnition 4.3.28. The core of a fuzzy subgroup µ of G, written core(µ) or µG , is deﬁned to be core(µ) = ∩x∈G µx Proposition 4.3.29. Let µ be a fuzzy subgroup of G. Then µG is a normal fuzzy subgroup of G. Proof. Since the intersection of any collection of fuzzy subgroups of G is a fuzzy subgroup of G, it suﬃces to show that µG is normal. Let z ∈ G. Then for all y ∈ G, µzG (y) = µG (z −1 yz) = ∩x∈G µx (z −1 yz) = ∩x∈G µ(x−1 z −1 yzx) = ∩x∈G µ((zx)−1 y(zx)) = ∩x∈G µx (y) = µG (y). Thus µG is a normal fuzzy subgroup of G. Proposition 4.3.30. If µ, ν are fuzzy subgroups of G, ν is normal and µ ⊇ ν, then µG ⊇ ν. Proof. The result follows since µx ⊇ ν x = ν for any x ∈ G and so µG = ∩x∈G µx ⊇ ν. Deﬁnition 4.3.31. The closure of a fuzzy subgroup µ of G, written µG , is deﬁned to be µG = {µx | x ∈ G}, the fuzzy subgroup generated by all the conjugates of µ in G.

4.3 Quasi-normal and Normal Fuzzy Subgroups

113

It follows that µG = ∪x∈G µx . The next result is immediate. Let Inn(G) denote the set of all inner automorphisms of G. Lemma 4.3.32. If µ is a fuzzy subgroup of G, then µG = ∩ν⊇f (µ) ν for all f ∈ Inn(G) and for all fuzzy subgroups ν of G. Proposition 4.3.33. Let µ be a fuzzy subgroup of G. Then µG is a normal fuzzy subgroup of G. Proof. It suﬃces to show that µG ⊇ g(µG ) for all g ∈ Inn(G). Let g ∈ Inn(G). Then for all f ∈ Inn(G) and for ν varying over F(G), we have that (Lemma 4.3.32) g(µG )(x) = g(∩ν⊇f (µ) ν)(x) = ∨{(∩ν⊇f (µ) ν)(z) | g(z) = x, z ∈ G} = ∨{∧{ν(z) | ν ⊇ f (µ), ν ∈ F(G)} | g(z) = x, z ∈ G} ≤ ∧{∨{ν(z) | g(z) = x, z ∈ G} | ν ⊇ f (µ), ν ∈ F(G)} ≤ ∩ν⊇f (µ) g(ν)(x) ≤ (∩g−1 (γ)⊇f (µ) γ)(x) (g(ν) = γ) ≤ ∩γ⊇g(f (µ)) γ(x) for all h ∈ InnG and γ ∈ F(G). ≤ ∩γ⊇h(µ) γ(x) ≤ µG (x). Therefore, µG is normal. If µ is a normal fuzzy subgroup of G, recall that the set G/µ = {xµ | x ∈ G} is a group with operation (xµ)(yµ) = (xy)µ. Lemma 4.3.34. If µ is a fuzzy subgroup of G and x, y ∈ G, then xµ = yµ if and only if µ(x−1 y) = µ(y −1 x) = µ(e). Proof. By Theorem 1.3.10, xµ = yµ ⇔ xµ∗ = yµ∗ ⇔ y −1 xµ∗ = eµ∗ = x−1 yµ∗ ⇔ y −1 xµ = eµ = x−1 yµ ⇔ µ(x−1 yz) = µ(z) = µ(y −1 xz) ∀z ∈ G ⇒ µ(x−1 y) = µ(e) = µ(y −1 x) ⇔ x−1 y, y −1 x ∈ µ∗ ⇔ xµ∗ = yµ∗ ⇔ xµ = yµ. Theorem 4.3.35. Let µ be a fuzzy subgroup of G. Then µ(x2 ) = µ(e) for all x ∈ G if and only if µ is normal and G/µ is an elementary Abelian 2 group. Proof. Suppose µ(x2 ) = µ(e) for all x ∈ G. Let x, y ∈ G. Then µ(e) = µ((xy)2 ) = µ(xyxy) = µ((xyx−1 )x2 y). Thus it follows by Lemma 3.1.5(2) that µ(xyx−1 ) = µ((x2 y)−1 ) = µ(y −1 (x2 )−1 ) µ(y −1 ) ∧ µ((x2 )−1 ) = µ(y −1 ) ∧ µ((x−1 )2 ) µ(y) ∧ µ(e) µ(y). Thus µ is a normal fuzzy subgroup. Hence G/µ is a group. Now by Lemma 4.3.34, it follows that the statement µ(x2 ) = µ(e) is equivalent to the statement x2 µ = eµ. Thus for all xµ in G/µ, (xµ)2 = (xµ)(xµ) = x2 µ = eµ. Hence every element in G/µ is of order 1 or 2. Therefore, G/µ is an elementary Abelian 2-group. The converse follows by Lemma 4.3.34.

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4 Characterization of Certain Groups and Fuzzy Subgroups

Corollary 4.3.36. Let µ be a fuzzy subgroup of G. Then µ(x2 ) = µ(e) for all x ∈ G if and only if µ is normal and G/µ∗ is an elementary abelian 2-group. Lemma 4.3.37. Let µ be a normal fuzzy subgroup of G and H a subgroup of G. Then the restriction of µ to H, µ|H , is a normal fuzzy subgroup of H. Moreover, H/µ|H is a subgroup of G/µ. Proof. It follows routinely that µ|H is a normal fuzzy subgroup of H. By Theorem 1.3.12(3), G/µ = G/µ∗ and H/µ|H =H/(µ| H )∗ . The natural homomorphism of G onto G/µ∗ maps H onto the set of left cosets {hµ∗ |h ∈ H}. Now (µ|H)∗ = H ∩ µ∗ and {hµ∗ |h ∈ H} is a group isomorphic to H/(H ∩ µ∗ ). Thus making the appropriate identiﬁcations, we can consider H/µ|H as a subgroup of G/µ. Lemma 4.3.38. If µ is a normal fuzzy subgroup of G, then any subgroup of G/µ can be written in the form H/µ|H , where H is a subgroup of G. Proof. Since any subset of G/µ is of the form H/µ|H , where H is a subset of G, it suﬃces to show that if H/µ|H is a subgroup of G/µ, then H is a subgroup of G. Let H/µ|H be a subgroup of G/µ and a, b ∈ H. Then aµ, bµ ∈ H/µ|H and so (aµ)(bµ)−1 ∈ H/µ|H . Thus ab−1 µ ∈ H/µ|H . Hence ab−1 ∈ H. Also, since eµ ∈ H/µ|H , e ∈ H. Therefore, H is a subgroup of G. In the remainder of the section, we write µ for µ|H when there is no chance for confusion. Theorem 4.3.39. Let µ be a normal fuzzy subgroup of G and H, K be subgroups of G. If H and K are conjugates, then H/µ and K/µ are conjugate in G/µ. Proof. Consider the function f : G → G/µ such that f (x) = xµ ∀x ∈ G. Then f is a homomorphism from G onto G/µ. By Lemma 4.3.26, H ∈ cl(K) implies H/µ ∈ f (cl(K)) = cl(f (K)) = cl(K/µ) since H, K are conjugate subgroups in G. Thus H/µ and K/µ are conjugate in G/µ. The converse of Theorem 4.3.39, is not true in general: Let G = D4 , the Dihedral group of degree 4, i.e., G = a, b | a4 = e, b2 = e, ba = a−1 b . Deﬁne the fuzzy subset µ of G as follows: ∀x ∈ G, t0 if x ∈ a, µ(x) = t1 otherwise, where 0 ≤ t1 < t0 ≤ 1. Clearly, µ is a normal fuzzy subgroup of G. Let H = {e, b, a2 , a2 b} and K = b . Then K/µ = {µ, bµ} and H/µ = {µ, bµ, a2 µ, a2 bµ}. Now t1 = µ(a2 e) = µ(ea2 ) = µ(e). Also, µ((a2 b)−1 b) = µ(ba2 b) = µ(b−1 a2 b) = t0 since b−1 a2 b ∈ a. By Lemma 4.3.34, a2 µ = µ and a2 bµ = bµ. Hence H/µ = {µ, bµ} = K/µ. Thus H/µ and K/µ are trivially conjugate in G/µ. However, K and H are not conjugate in G since they have diﬀerent orders.

4.3 Quasi-normal and Normal Fuzzy Subgroups

115

The above example shows that o(H/µ) = o(K/µ), but o(H) = o(K). Also, if L = a, then o(L) = o(H). However, o(L/µ) = o(H/µ) since o(H/µ) = 2 and o(L/µ) = 1. In general, we have the following result. Theorem 4.3.40. If G = D2n , the dihedral group of degree 2n , then there exists a normal fuzzy subgroup µ of G and subgroups H, K and L of G such that o(H) = o(K), but o(H/µ) = o(K/µ), o(K) = o(L), and o(K/µ) = o(L/µ). n Proof. Now G = a, b | a2 = e, b2 = e, ba = a−1 b . Deﬁne the fuzzy subset µ of G as follows: ∀x ∈ G, t0 if x ∈ a, µ(x) = t1 otherwise, where 0 ≤ t1 < t0 ≤ 1. Clearly, µ is a normal fuzzy subgroup of G. Let H = a and L = b . Now D2 n ∼ = Z2n−1 G, [17], and D2 n ∩ b = e , D2n b is a subgroup of G of order 2n . If we let K = D2 n b , then the desired result follows immediately. Theorem 4.3.41. Let µ be a normal fuzzy subgroup of G. Then the following conditions are equivalent: (1) G/µ is nilpotent of class at most n. (2) µ([x1 , x2 , ..., xn+1 ]) = µ(e) for all xi ∈ G, i = 1, 2, ..., n. (3) µ∗ ⊇ G(n) . Proof. (1) ⇒ (2) : G/µ is a nilpotent of class at most n ⇒ Zn (G/µ) = {eµ} (Theorem 4.3.4) ⇒ [x1 µ, x2 µ, ..., xn+1 µ] = eµ ⇒ [x1 , x2 , ..., xn+1 ]µ = eµ. Thus by Lemma 4.3.34, we have µ([x1 , x2 , ..., xn+1 ] = µ(e). (2) ⇒ (3) : The result here is clear. (3) ⇒ (1) : If µ∗ ⊇ G(n) , then µ([x1 , x2 , ..., xn+1 ]) = µ(e) which implies, by Lemma 4.3.34, that [x1 , x2 , ..., xn+1 ]µ = eµ. Thus [x1 µ, x2 µ, ..., xn+1 µ] = eµ and hence Zn (G/µ) = {eµ}. Thus G/µ is nilpotent of class at most n. As an immediate consequence of this theorem, we have the following result. Corollary 4.3.42. Let µ be a normal fuzzy subgroup of G. Then the following statements are equivalent. (1) G/µ is an Abelian group. (2) µ∗ ⊇ G(1) . (3) µ([x1 , x2 ]) = µ(e) for all xi ∈ G, i = 1, 2. We now characterize groups which have normal p-complements. Theorem 4.3.43. [14] Let p be a prime dividing o(G). Then G has a normal p-complement if and only if G contains no (p, q)-subgroup for all p, q ∈ π(G) with p = q.

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Theorem 4.3.44. Let p be an odd prime number dividing o(G). Then G has a normal p-complement if and only if there exists a fuzzy subgroup µ of G such that for all x ∈ G with o(x) = p, µ(x) is ﬁxed and µ(x) = µ([x, y]) for all y ∈ G\{e} such that p does not divide o(y). Proof. Suppose G has a normal p-complement. Then there exist subgroups P, K of G such that G = P K, K G, P ∩ K = e and P ∈ Sylp (G). Deﬁne a fuzzy subset  µ of G as follows:∀x ∈ G,  t0 if x = e, µ(x) = t1 if x ∈ K\e.  0 otherwise, where 0 < t1 < t0 ≤ 1. Clearly, µ is a fuzzy subgroup and [x, y] ∈ K for x ∈ P and y ∈ K. Thus µ([x, y]) t1 > 0 = µ(x). Hence µ([x, y]) = µ(x). Conversely, suppose G has no normal p-complement. Then by Theorem 4.3.43, G contains a (p, q)-subgroup H of G, p = q, p = 2. Hence by deﬁnition of H, H = P Q, where P ∈ Sylp (H) and Q ∈ Sylq (H). Furthermore, P ⊂ H, Q is cyclic, H = P and Exp(P ) = p. Thus for any x ∈ P and y ∈ Q, x = e = y, [x, y] ∈ P. Since H is not nilpotent, it follows that there exist x0 ∈ P and y0 ∈ Q, x0 = e = y0 , such that [x0 , y0 ] = e. Hence o(x0 ) = o([x0 , y0 ]) = p since expP = p. By the hypothesis µ(x0 ) = µ([x0 , y0 ]), a contradiction. The above theorem is not true in general if p = 2. For example, consider G = SL(2, 3). Then G is a (2, 3)-group and hence has no normal 2-complement. Now G has only one element of order 2, say z. Moreover, z = Z(G), [23]. Thus for all y ∈ G, [z, y] = e. Deﬁne a fuzzy subset µ of G as follows:  ∀x ∈ G,  t0 if x = e, t1 if x = z, µ(x) =  0 otherwise, where 0 < t1 < t0 ≤ 1. Then µ is a fuzzy subgroup of G and µ(z) = µ([z, y]). Thus we have a fuzzy subgroup µ of G which satisﬁes the hypothesis of Theorem 4.3.44, but G has no normal 2-complement. We now consider the case p = 2. Theorem 4.3.45. Let G be a group of even order. Then G has a normal 2complement if and only if there exists a fuzzy subgroup µ of G such that for all x, y ∈ G with o(x)|4, (o(x), o(y)) = 1 and µ(x) is ﬁxed implies µ(x) = µ([x, y]). Proof. The proof follows similarly as in the proof of Theorem 4.3.44 and by the fact that Exp(S) = 4, where S ∈ Syl2 (G). Theorem 4.3.46. Let G be a group of even order and S ∈ Syl2 (G). Then S is normal in G if and only if there exists a fuzzy subgroup µ of G such that µ(x) = µ([x, y]) and µ(x) = ﬁxed number for all x, y ∈ G, where o(x) is an odd prime and y is a 2-element.

References

117

Proof. Suppose S ∈ Syl2 (G) and S is normal. Then by Theorem 4.3.6, G contains no (p, 2)-subgroup for all p ∈ π(G)\{2}. Thus G = SK and (o(S), o(K)) = 1. Deﬁne a fuzzy subset µ of G as follows:  t0 if x = e, µ(x) = t1 if x ∈ S\e,  0 otherwise where 0 < t1 < t0 ≤ 1. Clearly, µ is a fuzzy subgroup and [x, y] ∈ S for all x ∈ K and y ∈ S. Therefore, µ([x, y]) t1 > 0 = µ(x). Hence µ([x, y]) = µ(x). Conversely, suppose G has no normal subgroup S, where S in Syl2 (G). Then by Theorem 4.3.6, G has a (p, 2)-subgroup H in G. Hence by deﬁnition of H, H = P Q, where P ∈ Sylp (L), P is normal in H, Q ∈ Syl2 (H) and p ≡ 1(2). Hence o(P ) = p else o(P/p ) = pn , n 2. However, from the structure of a (p, 2)-subgroup, n must be the smallest positive integer such that pn ≡ 1(2). Thus n = 1, a contradiction. Now for all x ∈ P and y ∈ Q, x = e = y, it follows that [x, y] ∈ P. Since H is not nilpotent, it follows that there exist x0 ∈ P and y0 ∈ Q, x0 = e = y0 , such that [x0 , y0 ] = e. Hence o([x0 , y0 ]) = o(x0 ) = p. Thus µ([x0 , y0 ]) = µ(x0 ), a contradiction.

References 1. S. Abou-Zaid, On normal fuzzy subgroups, J. Fac. Ed. Ain Shams Univ. Cairo 13(1988) 115-125. 2. S. Abou-Zaid, On generalized characteristic fuzzy subgroups of a ﬁnite group, Fuzzy Sets and Systems 43(1991) 235-241. 99 3. S. Abou-Zaid, On fuzzy subnormal and pronormal subgroup of a ﬁnite group, Fuzzy Sets and Systems 47(1992) 347-349. 4. S. Abou-Zaid, On fuzzy subgroups (short communication), Fuzzy Sets and Systems 55(1993) 237-240. 5. N. Ajmaal and K.V. Thomas, Quainormality and fuzzy subgroups, Fuzzy Sets and Systems 58 (1993) 217 - 225. 6. M. Akgul, Some properties of fuzzy groups, J. Math. Anal. Appl. 133(1988) 93-100. 107 7. M. Asaad, On the solvability, supersolvability, and nilpotency of ﬁnite groups, Annales Univ. Sci. Budapest XVI (1973) 115-124. 99 8. M. Asaad, Normal 2-subgroups of ﬁnite groups, Afrika Mat. 8(1986) 19-22. 105 9. M. Asaad, Groups and fuzzy subgroups, Fuzzy Sets and Systems 39(1991) 323328. 91 10. M. Asaad, Groups without certain minimal non-nilpotent subgroups, PU. M.A. Ser. A 2 (3-4) (1991) 157-160. 97, 99 11. M. Asaad and S. Abou-Zaid, Groups and fuzzy subgroups II (short communication), Fuzzy Sets and Systems 56 (1993) 375-377. 91 12. M. Asaad and S. Abou-Zaid, Fuzzy subgroups of nilpotent groups, Fuzzy Sets and Systems 60 (1993) 321-323. 91 13. M. Asaad and S. Abou-Zaid, Characterization of fuzzy subgroups, Fuzzy Sets and Systems 77 (1996) 247-251. 91 14. M. Asaad and S. Abou-Zaid, A contribution to the theory of fuzzy subgroups, Fuzzy Sets and Systems 77 (1996) 355-369. 91, 115

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15. P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264-269. 93, 94, 102, 103 16. M. Hall, The Theory of Groups, Macmillan, New York, 1959. 92, 94, 95 17. T. W. Hungerford, Algebra, Springer, New York, 1974. 97, 102, 115 18. B. Huppert, Endiche Gruppen I, Springer-Verlag, Berlin-Heidelberg-New York, 1967. 99 19. O. H. Kegel, Sylow-gruppen und subnormalteiler endlicher gruppen, Math. Z. 78 (1962) 205-221. 105 20. J-G Kim, On groups and fuzzy subgroups, Fuzzy Sets and Systems 67 (1994) 347 - 348. 21. R. Kumar, Fuzzy Sylow subgroups, Fuzzy Sets and Systems 46 (1992) 267 - 271. 22. W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133-139. 23. N. P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225-239. 116 24. J. S. Rose, A Course on Group Theory (Cambridge Univ. Press, Cambridge, 1978). 93, 105 25. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. 26. N. S. Narasimha Sastry and W. E. Deskins, Inﬂuence of normality conditions on almost minimal subgroups of a ﬁnite group, J. Algebra 52 (1978) 364-377. 106 27. W.R. Scott, Group Theory (Prentice-Hall, Englewood Cliﬀs, NJ, 1964). 93, 94, 98, 101, 111 28. F. I. Sidky and M.A. Mishref, Fuzzy cosets and cyclic abelian fuzzy subgroups, Fuzzy Sets and Systems 43 (1991) 243-250. 29. F. I. Sidky and M.A. Mishref, Fully invariant, characteristic and S-fuzzy subgroups, Inform. Sci. 55 (1991) 27-33. 99 30. L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353.

5 Free Fuzzy Subgroups and Fuzzy Subgroup Presentations

In this chapter, we deﬁne a notion of a free fuzzy subgroup and study its basic properties. We examine two approaches. The ﬁrst approach is based on the work appearing in [18]. It uses the notion of levels and leads to the development of the notion of a presentation of a fuzzy subgroup. The deﬁnition of a presentation for a fuzzy subgroup provides a convenient method of deﬁning fuzzy subgroups and opens the door for a development of a combinatorial group theory for fuzzy subgroups. The second approach follows along the lines of [28, 36]. The development here is presented in Section 5.3.

5.1 Free Fuzzy Subgroups An important concept in the study of many mathematical theories is that of a free object. For instance, free objects appear in abstract algebra [25, 26, 27], mathematical logic, lattice theory [19], category theory, and in universal algebra [17]. Free objects are also of importance in computer science. They appear in logic programming and automated theorem proving [14, 24], as the so called Herbrand Universe. They ﬁgure prominently in algebraic approaches to the semantics of programming languages [20]. Consequently, a suitable concept of freeness for fuzzy objects may prove useful in the study of fuzzy counterparts of these theories. It is worth noting that there is a strong connection between fuzzy set theory, fuzzy logic, and computer science. For example, work in this area can be found in [29] on fuzzy logic programming, [15] on fuzzy programming languages, and [23, 35] on fuzzy logic and automated theorem proving. In group theory, a notion related to that of a free object is the notion of a presentation. The notion of a presentation has proved useful in group theory. Presentations provide a convenient method of specifying deﬁning properties of a group. They have also proved to be interesting in the study of combinatorial group theory. Problems in combinatorial group theory often use ideas John Mordeson, Kiran R. Bhutani, Azriel Rosenfeld: Fuzzy Group Theory, StudFuzz 182, 119– 138 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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from group theory, logic, and theoretical computer science. These ﬁelds have beneﬁtted from the exchange of ideas. Let X be a set and let xt be a fuzzy singleton of X. We call x and t the foot and level of xt , respectively. Let Y be a set and f : X → Y. Let µ be a fuzzy subset of X. Recall that f (µ) is the fuzzy subset of Y deﬁned as follows: for all y ∈ Y, ∨{µ(x) | x ∈ f −1 (y)} if f −1 (y) = ∅, f (µ)(y) = 0 if f −1 (y) = ∅. If µ is a fuzzy subset of X and ν is a fuzzy subset of Y, then µ × ν is the fuzzy subset of X × Y deﬁned by for all x ∈ X, y ∈ Y, (µ × ν)(x, y) = µ(x) ∧ ν(y). We now recall some of the basic ideas concerning the construction of a free group. Let I be an index set. Let X = {xi | i ∈ I} be an alphabet of symbols | i ∈ I} be a set disjoint from X. Let Σ = X ∪X −1 and deﬁne and X −1 = {x−1 i ∗ Σ to be the set of all ﬁnite sequences of symbols from Σ, including the empty sequence. Σ is sometimes called a symmetrical alphabet. Elements from Σ ∗ are called words over Σ. Deﬁne the relation ∼ on Σ ∗ as follows: ∀w1 , w2 ∈ Σ ∗ , w1 ∼ w2 if and only if w1 can be transformed into w2 by a ﬁnite sequence of or x−1 insertions and/or deletions of subwords of the form xi x−1 i i xi , where ∗ i ∈ I. Then ∼ is an equivalence relation on Σ . Let [u] denote the equivalence class of u with respect to ∼, where u ∈ Σ ∗ . Let Σ ∗ / ∼ = {[u] | u ∈ Σ ∗ }. Deﬁne the binary operation on Σ ∗ / ∼ as follows: ∀[u], [v] ∈ Σ ∗ / ∼, [u][v] = [uv], where uv represents the concatenation of the words of u, v. Then Σ ∗ / ∼ forms a group called the free group over X, and is denoted by F (X). The identity of F (X) is the equivalence class determined by the empty word. Since F (X) = X, X is said to be the set of generators F (X). Each equivalence class [u] has a unique word in reduced form, i.e., one that contains no or x−1 subwords of the form xi x−1 i i xi . The following theorems are well known. Their proofs can be found in [25, 27]. Theorem 5.1.1. Every group is the homomorphic image of a free group. Theorem 5.1.2. Let a group G be generated by a set B = {gi | i ∈ I} and let X be an alphabet {xi | i ∈ I}. Then the function m : X → B deﬁned by ˆ : F (X) → G, in the m(xi ) = gi ∀xi ∈ X, extends to a unique epimorphism m sense that m([x]) ˆ = m(x) for all x ∈ X. The elements of the set X are called the deﬁning generators of the group G. By the ﬁrst isomorphism theorem for groups, G is isomorphic to F (X)/Ker(m). ˆ Hence G is completely determined (up to isomorphism) by specifying the alphabet X and the kernel of m. ˆ Ker(m) ˆ can itself be speciﬁed by giving a subset S of Ker(m) ˆ whose normal closure is Ker(m) ˆ and then by selecting a reduced word from each element of S. The set R of reduced words

5.1 Free Fuzzy Subgroups

121

obtained in this manner is called a set of deﬁning relators of G with respect to X. The pair X, R is called a presentation of G. We now consider the notion of a free fuzzy subgroup. First recall that xt ys = (xy)t∧s and (xt )−1 = (x−1 )t for all fuzzy singletons xt , ys of G. Deﬁnition 5.1.3. Let G, H be groups and let µ, ν be fuzzy subgroups of G, H, respectively. We say ν is a homomorphic image of µ if there exists an epimorphism h : G → H such that h(µ) = ν. If h is an isomorphism, then we say µ and ν are isomorphic. (See Deﬁnition 1.4.9.) Every w ∈ Σ ∗ can be written uniquely as w = xi1 xi2 ...xik with (j) = ±1, where x1 = x. We call the set {i1 , i2 , ..., ik }, the I-index set of w, and denote it by I(w). The inverse of the word w is the word (1) (2)

(k)

−(k) −(k−1) −(1) xik−1 ...xi1 .

w−1 = xik

We now deﬁne the concept of a free fuzzy subgroup. Deﬁnition 5.1.4. Let T = {ti ∈ [0, 1] | i ∈ I} and let t ∈ [0, 1] be such that t ∨{s | s ∈ T }. Deﬁne the fuzzy subset f (X; T, t) by, for all y ∈ F (X), f (X; T, t)(y) = ∨{∧{t ∧ ti | i ∈ I(w)} | w ∈ y}. X is called the set of generators, T is called the set of generating levels, and t is called the height of f (X; T, t). For all i ∈ I, ti is the level of xi and x−1 i . Theorem 5.1.5. The fuzzy subset f (X; T, t) of F (X) is a fuzzy subgroup of F (X). Proof. Set µ = f (X; T, t). Let y1 , y2 ∈ F (X) and let w1 ∈ y1 , w2 ∈ y2 . Then w1 w2 ∈ y1 y2 and µ(y1 y2 ) = ∨{∧{t ∧ ti | i ∈ I(w)}|w ∈ y1 y2 } ∧{t ∧ ti | i ∈ I(w1 w2 )} = ∧{∧{t ∧ tij | ij ∈ I(wj )}|j = 1, 2}. Since w1 , w2 are arbitrary, we have µ(y1 y2 ) ∨{∧{∧{t ∧ ti | i ∈ I(uj )}|j = 1, 2}|u1 ∈ y1 , u2 ∈ y2 } = (∨{∧{t ∧ ti |i ∈ I(u1 )}|u1 ∈ y1 }) ∧ (∨{∧{t ∧ ti |i ∈ I(u2 )}|u2 ∈ y2 }) = µ(y1 ) ∧ µ(y2 ). The proof that µ(y −1 ) ≥ µ(y) for all y ∈ G follows easily since for all y ∈ F (X), w ∈ y −1 if and only if w−1 ∈ y and I(w) = I(w−1 ). We call f (X; T, t) the free fuzzy subgroup of F (X) over X, T, and t.

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Theorem 5.1.6. Every fuzzy subgroup is the homomorphic image of a free fuzzy subgroup. Proof. Let G be a group with identity element e and µ a fuzzy subgroup of G. For all s ∈ µ(G), let Gs and Xs be sets and ms : Xs → Gs be a function such that the following conditions hold: (1) Gs is a set of generators for the level subgroup µs of G, (2) Xs is in one to one correspondence with Gs such that for all r ∈ µ(G) such that r = s, Xs ∩ Xr = ∅, (3) ms : Xs → Gs of (2) is a bijection. Set X = ∪s∈µ(G) Xs and deﬁne m : X → G as follows: ∀x ∈ X, m(x) = ms (x) if x ∈ Xs . Finally, deﬁne a function h : F (X) → G (1)

(k)

h([xi1 ...xik ]) =

k

m(xij )(j) ,

(5.1.1)

j=1

where the product on the right is the group operation of G. Then h is a homomorphism from F (X) onto G. Let T = {µ(m(xi )) | i ∈ I}, t = µ(e), and ν = f (X; T, t). We now show that h(ν) = µ. Let z ∈ G. By the deﬁnition of a free fuzzy subgroup, h(ν)(z) = ∨{∨{∧{µ(e) ∧ µ(m(xi )) | i ∈ I(w)}|w ∈ y}|h(y) = z}. (1)

(5.1.2) (k)

However, for all z ∈ G, y ∈ F (X) such that h(y) = z, and w = xi1 ...xik ∈ k (j) y, z = j=1 xij by Equation (5.1.1). Also, by the deﬁnition of a fuzzy subgroup, µ(z) ∧{µ(m(xij )) | j = 1, ..., k} = ∧{µ(m(xi )) | i ∈ I(w)} = ∧{µ(e) ∧ µ(m(xi )) | i ∈ I(w)}. By Equation (5.1.2), h(ν)(z) µ(z). We now consider the reverse inequality. There exist symbols xl1 , ..., xln in n Xµ(z) such that z = j=1 m(xlj )(j) in G. By the deﬁnition of Xµ(z) , ∧ {µ(e) ∧ µ(m(xli )) | i = 1, ..., n} µ(z). (1)

(5.1.3)

(n)

Set w = [xl1 ...xln ]. Then h(w) = z. It now follows from Equations (5.1.2) and (5.1.3) that h(ν)(z) ∧{µ(e) ∧ µ(m(xq )) | q ∈ I(w)} µ(z). Thus h(ν) = µ. The remainder of this section considers an extension of Theorem 5.1.2. The extension needs the concept of a generating set for the fuzzy case. For any set S, let SP = {xt | x ∈ S, t ∈ [0, 1]} be the set of all fuzzy singletons in S. If Q ⊆ SP , then the foot of Q is the set foot(Q) = {x ∈ S | xt ∈ Q}. Deﬁnition 5.1.7. Let ν be a fuzzy subgroup of G and let J ⊆ [0, 1]. A collection {Bi | i ∈ J} of nonempty subsets of GP is said to be generating for ν if the following conditions hold:

5.1 Free Fuzzy Subgroups

123

(1) ν(e) ∈ J and es(e) ∈ Bs(e) , where e is the identity element of G; (2) for every fuzzy singleton xt ∈ GP and each i ∈ J, xt ∈ Bi implies t = ν(x) = i; (3) for all x ∈ G, there exist ﬁnitely many x1 , ..., xk ∈ foot(∪i≥ν(x) Bi ) k (j) such that x = j=1 xj , where every (j) ∈ {−1, 1}. The last condition in the above deﬁnition is much stronger than the requirement that foot(∪i∈J Bi ) should generate G. Also, it is clear that for every fuzzy subgroup ν, the collection {Bi | i ∈ ν(G)} deﬁned by Bν(x) = {xν(x) | x ∈ G} is generating for ν. Thus every fuzzy subgroup has at least one such generating set. The next result follows from Deﬁnition 5.1.7. Lemma 5.1.8. Let ν be a fuzzy subgroup of G and let x ∈ G. If {Bi | i ∈ J} is generating for ν, then (1) there exist ﬁnitely many x1 , ..., xk ∈ foot(∪i∈J Bi ) such that k (j) x= xj and ν(x) = ν(x1 ) ∧ ... ∧ ν(xk ), j=1

(2) there exist ﬁnitely many fuzzy singletons (x1 )ν(x1 ) , ..., (xk )ν(xk ) ∈ ∪i∈J Bi such that k xν(x) = ((xj )ν(xj ) )(j) . j=1

Theorem 5.1.9. Let {Bi | i ∈ J} be a generating set for the fuzzy subgroup ν of G. Suppose the following conditions hold. (1) {Xi | i ∈ J} is a pairwise disjoint collection of sets of symbols such that for all i ∈ J, there exists a bijection mi : Xi → Bi . (2) X = ∪i∈J Xi , and m : X → G is the function deﬁned by m(x) = foot(mi (x)), where x ∈ Xi . Then there exists a unique epimorphism h : F (X) → G such that the following assertions hold. (3) h([x]) = m(x) for all x ∈ X, (4) h(f (X; T, t)) = ν, where T = {ν(m(x)) | x ∈ X} and t = ν(e). Proof. We may assume without loss of generality that X and T are indexed by the same set I. Deﬁne h : F (X) → G by h(w) =

k

m(xij )(j) ,

j=1 (1)

(k)

where w = [xi1 ...xik ] ∈ F (X). Then h is an epimorphism and clearly (3) holds. The uniqueness of h follows from Theorem 5.1.2.

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k (j) To prove (4), let z ∈ G. Then z = j=1 zj for some zj ∈ foot(∪i∈J B), (j) ∈ {−1, 1}, j = 1, ..., k. Using Lemma 5.1.8, we may assume that ν(z) = ∧{ν(zj ) | j = 1, ..., k}. By the deﬁnition of m, there exist xi1 , ..., xik ∈ X such that m(xij ) = zj (j) (k) for all j = 1, ..., k and h([xij ...xik ]) = z. Then, (f (X; T, t))(z) = ∨{∨{t ∧ ti | i ∈ I(w)|w ∈ u|h(u) = z}} (1)

(k)

∧{t ∧ ti | i ∈ I([xi1 ...xik ])} (1)

(k)

= ∧{ν(e) ∧ ν(m(xi )) | i ∈ I([xi1 ...xik ])} = ∧{ν(m(xij )) | j = 1, ..., k} = ∧{ν(zj ) | j = 1, ....k} = ν(g). We now consider reverse inequality. Let u ∈ F (X) and w ∈ u be such that n (1) (n) m(xij )(j) and so h(u) = z. Then w = xi1 ...xin . Thus z = h([w]) = j=1

ν(z) ∧{ν(m(xij )) | j = 1, ..., n} = ∧{ν(e) ∧ ν(m(xij )) | j = 1, ..., n} = ∧{t ∧ ti | i ∈ I(w)}. Since u, w are arbitrary, we get ν(z) ∨{∨{∧{t ∧ ti | i ∈ I(w)} | w ∈ u} | h(u) = z} = h(f (X; T, t))(z) and so the desired inequality holds. We note that in the free fuzzy subgroup f (X; T, t) of Theorem 5.1.9, the level of each x in Xi equals i. Thus knowing {Xi | i ∈ J} renders T superﬂuous. Further, since eν(e) ∈ B(ν(e)), speciﬁcation of t = ν(e) is also unnecessary. Since X = ∪i∈J Xi , X, T, and t can all be recovered from the knowledge of {Xi | i ∈ J}. We therefore simplify the notation f (X; T, t) to f ({Xi | i ∈ J}).

5.2 Presentations of Fuzzy Subgroups The notion of a quotient is needed in order to deﬁne a presentation. The notion of a quotient together with the results of the previous section are used to extend the concept of a presentation to fuzzy subgroups. Let µ be a fuzzy subgroup of G and let N be a normal subgroup of G. Deﬁne the fuzzy subset µN of G/N by for all x ∈ G/N, µN (x) = ∨{µ(y) | y ∈ xN }. Then µN is a fuzzy subgroup of G/N by Theorem 1.3.13. We call µN the quotient of µ by N. Let ν be a fuzzy subgroup G with {Bi | i ∈ J} its generating set. There exists a collection {Xi | i ∈ J} of sets of symbols and a collection of functions

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{mi :Xi → Bi | i ∈ J and (1) and (2) of Theorem 5.1.9 hold}. Let X = ∪i∈J Xi and let h : F (X) → G be an epimorphism satisfying (3) and (4) of Theorem 5.1.9. Set µ = µ((Xi )i∈J ), N = ker(h), and consider the quotient µN of µ by N. Deﬁne ψ : F (X)/N → G by for all y ∈ F (X), ψ([y]) = h(y). Clearly ψ is an isomorphism. Let z ∈ G. Then ψ(µN )(z) = ∨{fN (u) | ψ(u) = z} = ∨{µ(w) | w ∈ u, ψ(u) = z} = ∨{µ(w) | ψ([w]) = z} = h(µ)(z). Since h(µ) = ν by Theorem 5.1.9, it follows that µN and ν are isomorphic. From the above discussion, we see that to specify the fuzzy subgroup ν of G (up to isomorphism), it suﬃces to specify a collection {Xi | i ∈ J} of sets of symbols with each Xi being in one to one correspondence with Bi for some collection {Bi | i ∈ J} generating ν, as well as the subgroup N = ker(h) of F (X), where X = ∪i∈J Xi . N may be speciﬁed using a subset S of N whose normal closure is N. Every element eS of S may be speciﬁed by selecting the reduced word in eS as its representative. Let R be the set of all such reduced words. Then ν is speciﬁed by the pair {Xi | i ∈ J}, R . This pair is called a presentation of ν. The collection {Xi | i ∈ J} is called the generating collection, and R is called the set of relators of ν with respect to {Xi | i ∈ J}. Since the domain of µN , F (X)/N, is isomorphic to the domain of ν, G, the presentation of the fuzzy subgroup ν of G yields a presentation for the domain of ν. Moreover, the two presentations have the same relators. We now give two examples to illustrate the deﬁnitions and main theorems. Example 5.2.1. Deﬁne the fuzzy subgroup ρ of Z12 as follows: ∀x ∈ Z12 ,   1 if x = 0 or x = 6, ρ(x) = 12 if x = 3 or x = 9, 1 otherwise. 3 To obtain a presentation for ρ, we form the generating collection {Bi | i ∈ J}, where J = {1, 12 , 13 }, and a collection {Xi | i ∈ J} as in Theorem 5.1.9: B1 = {01 , 61 }, X1 = {z, s}, B 21 = {3 12 }, X 21 = {t}, B 31 = {1 13 }, X 31 = {u}. Here z, s, t and u stand for zero, six, three, and one, respectively. For every i ∈ J, mi : Xi → Bi is the obvious bijection. In particular, m1 (z) = 01 , m1 (s) = 61 , m 12 (t) = 3 12 , and m 31 (u) = 1 13 . Let F (X) be the free group on X = ∪i∈J Xi , where X is assumed to be indexed by itself. The free fuzzy subgroup µ of F (X)

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is given by µ(y) = ∧{t ∧ ti | i ∈ X(w)}, where w is the reduced word in the equivalence class y ∈ F (X) and t = 1 = ∨J . We identify every such y with its reduced word for the sake of convenience. We now give some examples of computing the values of the fuzzy free subgroup µ at some points. For example, 1 1 = , 3 3 1 1 1 1 µ(utst−1 ) = 1 ∧ ∧ ∧ 1 ∧ = , 3 2 2 3 1 1 1 µ(tt) = 1 ∧ ∧ = . 2 2 2 The value of µ at the empty word is 1 = ∨J. The collection {Xi | i ∈ J} constitutes the generators of the presentation. By Theorem 5.1.9, there exists a unique homomorphism h such that F (X)/ ker(h) Z12 . The kernel of h gives the relators of the (fuzzy) presentation. Condition (3) of Theorem 5.1.9 yields the relations h(z) = 0, h(s) = 6, h(t) = 3, h(u) = 1. From these it follows that the deﬁning relations are s = t2 , t = u3 . z = e, s2 = e, Thus the fuzzy subgroup ρ has the presentation z1 , s1 , t 12 , u 13 | z = e, s2 = e, s = t2 , t = u3 . µ(u) = 1 ∧

Example 5.2.2. Consider the dihedral group D4 = a, b | a4 = e, b2 = e, ba = a3 b . Let ρ bethe fuzzy group of D4 deﬁned as follows: 1 if x = e,   1 if x = a2 b, ρ(x) = 21 if x ∈ {a2 , b},    31 if x ∈ {a, a3 , ab, ba}. 4 Then B1 = {e1 }, B 21 = {w 12 },

X1 = {e} X 12 = {w},

B 31 = {x 13 , y 13 },

X 13 = {x, y}

B 41 = {z 14 },

X 14 = {z},

where w = a2 b, x = a2 , y = b, and z = a. It follows that there is a unique homomorphism h satisfying the relations h(e) = e,

h(w) = a2 b,

h(y) = b,

h(z) = a.

h(x) = a2 ,

This yields the presentation e1 , w 12 , x 13 , y 13 , z 14 | w = xy, z 4 = e, y 2 = e, x = z 2 , yz = z 3 y , where the overlines on the generators of the presentation are suppressed.

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127

5.3 Constructing Free Fuzzy Subgroups In this section, we give an outline of another approach for the construction of free fuzzy subgroups. The approach diﬀers from that in Section 5.1. It follows along the lines of [28] and [36]. If X is a set and χ a fuzzy subset of X, we call (X, χ) a fuzzy set in this section. Let G be a group and µ a fuzzy subgroup of G. We call (G, µ) a fuzzy group in this section. Let X be a subset of G and let χ be a fuzzy subset of X such that χ ⊆ µ. We call (X, χ) a fuzzy subset of (G, µ). We say that (X, χ) generates (G, µ) if G = X and µ = χ, where we consider χ to be the fuzzy subset of G obtained by extending χ from X to G such that χ(x) = 0 for all x ∈ G\X. If (X, χ) generates (G, µ), we write (G, µ) = X, χ. Let (X, µ) and (Y, ν) be fuzzy sets and f a function of X into Y. If f (µ) ⊆ ν, then f is called a fuzzy function of (X, µ) into (Y, ν). If f (X) = Y and f (µ) = ν, then f is called surjective. Let (G, µ) and (H, ν) be fuzzy groups and let f be a fuzzy function of (G, µ) into (H, ν) such that f is a homomorphism of G into H. If f is an epimorphism (isomorphism) of G onto H and f (µ) = ν, then f is called an epimorphism (isomorphism) of (G, µ) onto (H, ν). Deﬁnition 5.3.1. Let (F, µ) be a fuzzy group and (X, χ) a fuzzy set, where X ⊆ F. Then (F, µ) is called a free fuzzy group on (X, χ) provided the following conditions hold. (1) (X, χ) generates (F, µ). (2) If (G, ν) is any fuzzy group having a generating fuzzy set (Y, η), then for any surjection f : (X, χ) → (Y, η), there exists an epimorphism f ∗ : (F, µ) → (G, ν) such that f ∗ (x) = f (x) for all x ∈ X. We call the fuzzy set (X, χ) in Deﬁnition 5.3.1 a free fuzzy basis for the group (F, µ). The proof of the next result follows from the known situation for free groups and from Deﬁnition 5.3.1. We note from Deﬁnition 5.3.1 that if f ∗ : (F, µ) → (G, ν) is an epimorphism, then f ∗ (µ) = ν. Theorem 5.3.2. Let (Fi , µi ) be free fuzzy groups on fuzzy sets (Xi , χi ) with i = 1, 2, respectively. Suppose f1 is a one-to-one function of X1 onto X2 such that f1 (χ1 ) = χ2. Then there is an isomorphism of fuzzy groups from (F1 , µ1 ) onto (F2 , µ2 ). We now show that for any fuzzy set (X, χ), there exists a free fuzzy group on (X, χ). Results on free groups can be found in [16, 25, 27]. Let Σ = X ∪ X −1 and Σ ∗ be as in Section 5.1. Let F (X) = Σ ∗ / ∼ as in Section 5.1. We write X ± = X ∪ X −1 . Let χ be a fuzzy subset of X and χ−1 the fuzzy subset of X −1 such that χ−1 (x−1 ) = χ(x) for all x ∈ X. A word is a ﬁnite sequence of symbols, w = xε11 ...xεnn , n ≥ 0, εi = ±1, xi ∈ X for i = 1, ..., n, where n ∈ N. If n = 0, then w = e, the empty word. Recall that the set Σ ∗ of all words is a monoid under concatenation.

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Let w = xε11 ...xεnn ∈ Σ ∗ with xi ∈ X, εi = ±1, i = 1, ..., n. We call n the length of w and denote it by |w|. If e is the empty string, then e has length 0 and we write |e| = 0. Deﬁne the fuzzy subset ξ of Σ ∗ as follows: ∀ w = xε11 ...xεnn ∈ Σ ∗ , ξ(w) = ∧{χ(xi ) | 0 ≤ i ≤ |w|}. Then the following proposition holds. Proposition 5.3.3. If the fuzzy set (Σ ∗ , ξ) is as above, then ξ is a fuzzy subgroupoid of Σ ∗ . Proposition 5.3.4. Let µ be a fuzzy subset of a group G. Then µ(x) = ∨{r|x ∈ µr , r < ∨µ}. Proof. Let µ∗ be the fuzzy subset of G deﬁned by ∀x ∈ G, µ∗ (x) = ∨{r|x ∈ µr , r < ∨µ}. Suppose there exist x, y ∈ G such that µ∗ (xy) < µ∗ (x) ∧ µ∗ (y). Then µ∗ (x) > µ∗ (xy) and µ∗ (y) > µ∗ (xy). Thus there exist t1 , t2 ∈ [0, 1] such that t1 , t2 > µ∗ (xy) and x ∈ µt1 , y ∈ µt2 . Suppose t2 ≥ t1 . Then x, y ∈ µt2 ⊆ µt1 . Since t1 > µ∗ (xy) = ∨{t|xy ∈ µt , t ≤ ∨µ}, we have that xy ∈ / µt1 . However, this contradicts the fact that µt1 is a subgroup of G. Since x ∈ µt ⇔ x−1 ∈ µt for all x ∈ G, it follows that µ∗ (x) = µ∗ (x−1 ) for all x ∈ G. Thus µ∗ is a fuzzy subgroup of G. Let x ∈ G and let µ(x) = t. Then x ∈ µt . Thus µ∗ (x) ≥ t and so µ ⊆ µ∗ . Therefore, µ ⊆ µ∗ . we have equality once we show that µ∗ is the smallest fuzzy subgroup of G containing µ. Let θ be any fuzzy subgroup of G containing µ. Suppose that µ∗ (x) > θ(x) for some x ∈ G. Then there exists t ∈ [0, 1] such that t > (x) and x ∈ µt . Thus x = x1 x2 ...xn for some xi ∈ µt such that µ(xi ) ≥ t or −1 ) µ(x−1 i ) ≥ t, i = 1, 2, ..., n. Since θ is a fuzzy subgroup of G, θ(xi ) = θ(x ) ≥ t for i = 1, 2, ..., n. Thus for i = 1, 2, ..., n. Hence θ(xi ) ≥ µ(xi ) ∨ µ(x−1 i θ(x) = θ(x1 x2 ...xn ) ≥ θ(x1 ) ∧ θ(x2 ) ∧ ... ∧ θ(xn ) ≥ t. However, this contradicts the fact that θ(x) < t. Thus µ∗ ⊆ θ. −ε1 n We deﬁne the inverse w−1 of w = xε11 ...xεnn to be w−1 = x−ε n ...x1 , −1 (wv) = v −1 w−1 and e−1 = e. An elementary transformation of a word w consists of inserting or deleting a part of the form xx−1 for x ∈ X ±1 . Two words w and v are equivalent, w ∼ v, if there is a chain of elementary transformations leading from w to v. Then the relation ∼ is an equivalence relation on W. We let [w] denote the equivalence class of w ∈ X ± . The product of equivalence classes [w] and [v] is deﬁned by [w] [v] = [wv] . This product is well deﬁned. This F (X) = Σ ∗ / ∼ is a group. Deﬁne g : Σ ∗ → F (X) by ∀w ∈ Σ ∗ , g(w) = [w]. Then g is a homomorphism of Σ ∗ onto F (X). The image of ξ under g is as follows: ∀[w] ∈ F (X),

g(ξ)([w]) = ∨{ξ(u) | g(u) = [w], u ∈ Σ ∗ } = ∨{∧{χ(xi ) | 0 ≤ i ≤ n}| u = x11 ...xnn , g(u) = [w], u ∈ Σ ∗ }. Let µ = g(ξ). Then it follows that µ is a fuzzy subgroup of F (X).

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We now show that the fuzzy group (F (X), µ) above is a free fuzzy group on (X, χ). We ﬁrst show that (F (X), µ) = X, χ. Since the function sending x to [x] for x ∈ X is clearly one-one, X can be regarded as a subset of F (X) when x is identiﬁed with [x]. Thus F (X) = X. Next we show that µ = χ, where for all [w] ∈ F (X), µ([w]) = ∨{∧{χ(xi ) | 0 ≤ i ≤ n}| u = x11 ...xnn , g(u) = [w], u ∈ Σ ∗ } and χ([w]) = ∨{k | [w] ∈ χk , k < ∨χ}. by Proposition 5.3.4. For all [w] ∈ F (X), g(w) = [w] with w ∈ Σ ∗ . Let t = χ([w]). Then it follows that [w] ∈ χt . Since χt = χt− for any suﬃciently small positive number , it follows that [w] ∈ χt = χt− . Since there exist x1 , ..., xn ∈ χt− such that w = xε11 ...xεnn , where n ≥ 0 and εi = ±1, it follows that χ (xi ) ≥ t − . Hence ∧{χ (xi ) | i = 1, ..., n} ≥ t − , where w = xε11 ...xεnn . Thus it follows that µ([w]) = ∨{∧{χ(xi ) | 0 ≤ i ≤ n}| g(u) = [w], u = x11 ...xnn , u ∈ Σ ∗ } ≥ t = χ([w]). We now show that µ([w]) ≤ χ([w]). For all [w] ∈ F (X), it follows that µ([w]) = ∨{∧{χ(xi ) | 0 ≤ i ≤ n}| u = x11 ...xnn , g(u) = [w], u ∈ Σ ∗ } = ∨{k | k = ∧{χ(xi ) | 0 ≤ i ≤ n}| g(u) = [w] , u = xε11 ...xεnn , u ∈ Σ ∗ }. Now w ∈ χk implies that there exist xi ∈ χk such that w = xε11 ...xεnn , where n ∈ N and εi = ±1,and χ(xi ) ≥ k for all xi with i = 1, ..., n. Since ∧{χ(xi ) | 0 ≤ i ≤ n} ≤ χ(xi ), it follows that µ([w]) = ∨{k | k = ∧{χ(xi ) | 0 ≤ i ≤ n}| g(u) = [w] , u = xε11 ...xεnn , u ∈ Σ ∗ } ≤ ∨{k | [w] ∈ χk , k < ∨χ} = χ([w]). Hence µ([w]) ≤ χ([w]). Therefore, µ = χ and so (F, µ) = X, χ . We now consider condition (2) of Deﬁnition 5.3.1. Let (G, ν) be a fuzzy group having a generating fuzzy set (Y, η) such that f : (X, χ) → (Y, η) is surjective. Then by setting f (x−1 ) = f (x)−1 with x ∈ X, ϕ induces a function of X ±1 onto Y ±1 . Deﬁne an extension f of f from Σ ∗ onto G by ∀w = xε11 ...xεnn , f (w) = f (x1 )ε1 ...f (xn )εn , xi ∈ X, εi = ±1 for i = 1, ..., n. If w1 and w2 are equivalent, then f (w1 ) = f (w2 ) and so f maps equivalent words onto the same element of G. Thus f induces a function f ∗ : F (X) → G deﬁned by f ∗ ([w]) = f (w). Clearly, f ∗ is an epimorphism of groups. It remains to show that f ∗ is an epimorphism in a fuzzy sense, i.e., f ∗ (µ) = ν. Suppose that (G, ν) = Y, η and that f : (X, χ) → (Y, η) is surjective, where Y ⊆ G. Then ν is the fuzzy subgroup of G generated by η and f (χ) = η. Thus it suﬃces to show that f ∗ (µ) = ν, that is, to show that

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∨{∨{k | [w] ∈ χk , k < ∨χ}| f ∗ ([w]) = w} = ∨{k | w ¯ ∈ ηk , k < ∨η} for all w ¯ ∈ G. Let w ¯ ∈ G and t = ν(w). ¯ Then for all suﬃciently small positive number , it follows that w ¯ ∈ νt = ηt− . Hence there exist yi ∈ ηt− such that w ¯ = y1ε1 ...ynεn , n ≥ 0, εi = ±1. Thus η(yi ) ≥ t − . For all yi ∈ Y with w ¯ = y1ε1 ...ynεn , there exist xi ∈ X such that ϕ(xi ) = yi and w = xε11 ...xεnn since f is surjective. Furthermore, it follows that ∨{χ(xi ) | f (xi ) = yi } = η(yi ) for all yi = Y ±1 . Thus ∨{k | w ∈ χk } ≥ t − . Therefore, it follows that ¯ ∈ G. ∨{∨{k | [w] ∈ χk , k < ∨χ} | f ∗ ([w]) = w} ≥ t for all w Hence f ∗ (µ) ⊇ ν. We now show that f ∗ (µ) ⊆ ν. Since f ∗ is an epimorphism of groups, for w ¯ ∈ G there exist [w] ∈ F (X) ¯ Set t = µ([w]) = ∨{k | [w] ∈ χk , k < ∨χ}. Then such that f ∗ ([w]) = w. it follows that [w] ∈ µt = χt− for suﬃciently small positive number . Hence there exist xi ∈ χt− such that w = xε11 ...xεnn , n ∈ N, εi = ±1 and χ(xi ) ≥ t − , i = 1, ..., n. Since f is a fuzzy function, it follows that χ(x) ≤ η(f (x)) for all x ∈ X. Hence t − ≤ χ(xi ) ≤ η(f (xi )) with w = xε11 ...xεnn . Since f (x1 )ε1 ...f (xn )εn = ¯ ∈ ηk }. w ¯ and is arbitrary, it follows that t ≤ f (ϕ(xi )). Thus t ≤ ∨{k | w ¯ it follows that Therefore, since [w] ∈ F (X) is arbitrary and f ∗ ([w]) = w, ¯ = ∨{∨{k | [u] ∈ χk , k < ∨χ}| f ∗ ([u]) = w} (f ∗ (µ))(w) ≤ ∨{k | w ¯ ∈ ηk , k < ∨η} = ν(w). ¯ By the preceding inequalities, it follows that f ∗ (µ) = ν. We thus have the following result. Theorem 5.3.5. Every fuzzy group is a homomorphic image of a free fuzzy group. The interested reader can ﬁnd results on free products of fuzzy subgroups in [1] and [33].

5.4 Free (s,t]-Fuzzy Subgroups In [8, 10, 30] the idea of a fuzzy point and its membership to and quasicoincidence with a fuzzy set were used to deﬁne and study certain kinds of fuzzy topological spaces and (∈, ∈ ∨q)-fuzzy subgroups, respectively. Let G be a group. The purpose of this section is two-fold. First, we wish to extend

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131

the notion of an (, ∨ q)-fuzzy subgroup of G introduced in [8, 10] to a more general situation. It is shown in [10] that µ is an (, ∨ q)-fuzzy subgroup of G if and only if ∀x, y ∈ G, µ(xy −1 ) ≥ µ(x) ∧ µ(y) ∧ 12 . Let c : [0, 1] → [0, 1] . Then c is called a fuzzy complement if (1) c(0) = 1 and c(1) = 0 and (2) ∀s, t ∈ [0, 1], s ≤ t ⇒ c(s) ≥ c(t), [[22], p.52]. Let c be a fuzzy complement and let α ∈ [0, 1]. If c(α) = α, then α is called an equilibrium of c and c is said to have an equilibrium. It is known that if c has an equilibrium, then the equilibrium is unique [[22],Theorem 3.2, p. 57]. Also, ∀s, t ∈ [0, 1], t > s and t > c(s) ⇒ t > α. We assume throughout that c has an equilibrium α, i.e., there is an element α ∈ [0, 1] such that c(α) = α. We deﬁne a c-quasi-fuzzy subgroup and show that µ is a c-quasi-fuzzy subgroup of G if and only if ∀x, y ∈ G, µ(xy −1 ) ≥ µ(x) ∧ µ(y) ∧ α. If c is is the standard fuzzy complement deﬁned by c(x) = 1 − x ∀x ∈ [0, 1], then α = 12 . Hence we have the extension referred to above. Second, we study the notion of an (s, t]-fuzzy subgroup as introduced in [40]. We feel that this notion has possible applications for suitable choices of s and t in certain situations. The notion of an (s, t]-fuzzy subgroup is more general than that of a c-quasi-fuzzy subgroup. The latter can be obtained by letting s = 0. We present some basic results of (s, t]-fuzzy subgroups before considering the notion of a free (s, t]-fuzzy subgroups [13]. Many basic results concerning the above types of fuzzy subgroups have been presented in [2] [8], [11, 21], [37] - [40]. Throughout this section, we assume that c is a fuzzy complement and that c has an equilibrium α. We also assume that G is a group. Deﬁnition 5.4.1. Let µ be a fuzzy subset of G. Then µ is called a c-quasifuzzy subgroup of G if ∀x, y ∈ G and ∀t, r ∈ [0, 1], xt , yr ⊆ µ implies (xy −1 )t∧r ⊆ µ or t ∧ r > (cµ)(xy −1 ). Theorem 5.4.2. Let µ be a fuzzy subset of G. Then µ is a c-quasi-fuzzy subgroup of G if and only if ∀x, y ∈ G, µ(xy −1 ) ≥ µ(x) ∧ µ(y) ∧ α. Proof. Suppose µ is a c-quasi fuzzy subgroup of G. Suppose ∃x, y ∈ G such that µ(xy −1 ) < µ(x) ∧ µ(y) ∧ α. Then µ(xy −1 ) < α. Let xt0 , yr0 ⊆ µ. Suppose t0 ∧ r0 > µ(xy −1 ). Then by hypothesis, t0 ∧ r0 > (cµ)(xy −1 ). Hence t0 ∧ r0 > α. That is, µ(xy −1 ) < α < t0 ∧ r0 . Let t ≤ t0 and r ≤ r0 be such that µ(xy −1 ) < t ∧ r < α. Now xt , yr ⊆ µ and by an argument as above, t ∧ r > α, a contradiction. Hence either µ(xy −1 ) ≥ µ(x) ∧ µ(y) ∧ α, the desired result, or t0 ∧ r0 ≤ µ(xy −1 ). In the latter case, we can take t0 = µ(x) and r0 = µ(y) and so µ(xy −1 ) ≥ µ(x) ∧ µ(y). Conversely, suppose ∀x, y ∈ G, µ(xy −1 ) ≥ µ(x)∧µ(y)∧α. Suppose xt , yr ⊆ µ. Suppose µ(xy −1 ) < t∧r. Then it suﬃces to show t∧r > (cµ)(xy −1 ). Suppose t ∧ r ≤ (cµ)(xy −1 ). Then µ(xy −1 ) < t ∧ r ≤ (cµ)(xy −1 ). Hence µ(xy −1 ) < (cµ)(xy −1 ) and so µ(xy −1 ) < α. Thus µ(xy −1 ) ≥ µ(x) ∧ µ(y) (≥ t ∧ r) since µ(xy −1 ) ≥ µ(x) ∧ µ(y) ∧ α . Hence µ(xy −1 ) ≥ t ∧ r. Thus µ(xy −1 ) < t ∧ r and µ(xy −1 ) ≥ t ∧ r, a contradiction. Therefore, t ∧ r > (cµ)(xy −1 ).

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Proposition 5.4.3. Suppose µ is a c-quasi-fuzzy subgroup of G. Let x, y ∈ G. If µ(x) > µ(y), then µ(xy) ∧ α = µ(y) ∧ α. Proof. µ(y) = µ(x−1 xy) ≥ µ(x−1 ) ∧ µ(xy) ∧ α ≥ µ(x) ∧ α ∧ µ(xy) ∧ α = µ(x) ∧ µ(xy) ∧ α ≥ µ(x) ∧ µ(x) ∧ µ(y) ∧ α ∧ α = µ(y) ∧ α. Thus µ(y)∧α ≥ µ(x)∧µ(xy)∧α ≥ µ(y)∧α. Thus µ(y)∧α = µ(x)∧µ(xy)∧α. Since µ(x) > µ(y) ∧ α, µ(y) ∧ α = µ(xy) ∧ α. Proposition 5.4.4. Suppose µ is an c-quasi-fuzzy subgroup of G. Let x, y ∈ G. If µ(x) ∧ α > µ(y) ∧ α, then µ(xy) ∧ α = µ(y) ∧ α = µ(y) = µ(xy). Proof. As in the proof of the previous proposition, µ(y)∧α = µ(x)∧µ(xy)∧α. Since µ(x) ∧ α > µ(y) ∧ α, µ(y) ∧ α = µ(xy). Now µ(x) ∧ α > µ(y) ∧ α implies α ≥ µ(y). Thus µ(y) = µ(xy). Deﬁnition 5.4.5. [40] Let G be a group and µ a fuzzy subset of G. Then for s, t ∈ [0, 1] , s < t, µ is called an (s, t]-fuzzy subgroup of G if the following conditions hold: (1) ∀x, y ∈ G, µ(xy) ∨ s ≥ µ(x) ∧ µ(y) ∧ t, (2) ∀x ∈ G, µ(x−1 ) ∨ s ≥ µ(x) ∧ t. Theorem 5.4.6. [40] Let µ be a fuzzy subset of G. Then µ is an (s, t]-fuzzy subgroup of G if and only if µr is a subgroup of G for all r ∈ (s, t]. Theorem 5.4.7. Let f be a homomorphism of a group G into a group K. (1) If µ is an (s, t]-fuzzy subgroup of G, then f (µ) is an (s, t]-fuzzy subgroup of K. (2) If ν is an (s, t]-fuzzy subgroup of K, then f −1 (ν) is an (s, t]-fuzzy subgroup of G. Proof. (1) Let y1 , y2 ∈ f (G). Then f (µ)(y1 y2 ) ∨ s = ∨{µ(x) | f (x) = y1 y2 , x ∈ G} ∨ s ≥ ∨{µ(x1 x2 ) | f (x1 ) = y1 , f (x2 ) = y2 , x1 , x2 ∈ G} ∨ s = ∨{µ(x1 x2 ) ∨ s | f (x1 ) = y1 , f (x2 ) = y2 , x1 , x2 ∈ G} ≥ ∨{µ(x1 ) ∧ µ(x2 ) ∧ t | f (x1 ) = y1 , f (x2 ) = y2 , x1 , x2 ∈ G} = ∨{µ(x1 ) ∧ t | f (x1 ) = y1 , x1 ∈ G} ∧ ∨{µ(x2 ) ∧ t | f (x2 ) = y2 , x2 ∈ G} = ∨{µ(x1 ) | f (x1 ) = y1 , x1 ∈ G} ∧ ∨{µ(x2 ) | f (x2 ) = y2 , x2 ∈ G} ∧ t = f (µ)(y1 ) ∧ f (µ)(y2 ) ∧ t. / f (G) or y2 ∈ / f (G). Then either f (µ)(y1 ) = 0 or Suppose either y1 ∈ f (µ)(y2 ) = 0. Hence f (µ)(y1 y2 ) ∨ s = 0 = f (µ)(y1 ) ∧ f (µ)(y2 ) ∧ t. Let y ∈ f (G). Then f (µ)(y −1 ) ∨ s = ∨{µ(x−1 ) | f (x−1 ) = y −1 , x−1 ∈ G} ∨ s = ∨{µ(x−1 ) ∨ s | f (x−1 ) = y −1 , x−1 ∈ G} ≥ ∨{µ(x) ∧ t | f (x−1 ) = y −1 , x−1 ∈ G} = ∨{µ(x) | f (x−1 ) = y −1 , x−1 ∈ G} ∧ t = ∨{µ(x) | f (x) = y, x ∈ G} ∧ t = f (µ)(x) ∧ t. / f (G). Hence f (µ)(y −1 ) = 0 = f (µ)(y). Suppose y ∈ / f (G). Thus y −1 ∈ −1 Thus f (µ)(y ) ∨ s ≥ f (µ)(y) ∧ t. (2) Let x1 , x2 ∈ G. Then f −1 (ν)(x1 x2 ) ∨ s = ν(f (x1 x2 )) ∨ s = ν(f (x1 )f (x2 )) ∨ s ≥ ν(f (x1 ) ∧ ν(f (x2 )) ∧ t = f −1 (ν)(x1 ) ∧ f −1 (ν)(x2 ) ∧ t. Let

5.4 Free (s,t]-Fuzzy Subgroups

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x ∈ G. Then f −1 (ν)(x−1 ) ∨ s = ν(f (x−1 )) ∨ s = ν(f (x)−1 ) ∨ s ≥ ν(f (x)) ∧ t = f −1 (ν)(x) ∧ t. Proposition 5.4.8. Suppose G is a group and s, t ∈ [0, 1] with s < t. Let H be a subgroup of G. Suppose µ is a fuzzy subset of G such that ∀x ∈ G, µ(x) ≥ t if x ∈ H and µ(x) ≤ s if x ∈ / H. Then µ is an (s, t]-fuzzy subgroup of G. Proof. Let x, y ∈ G. Suppose x, y ∈ H. Then µ(xy) ∨ s = µ(xy) ≥ t = µ(x) ∧ µ(y) ∧ t. Also µ(x−1 ) ∨ s = µ(x−1 ) ≥ t = µ(x) ∧ t. Suppose x ∈ H and y ∈ / H. Then xy ∈ / H. Hence µ(xy) ∨ s = s ≥ µ(x) ∧ µ(y) ∧ t since µ(y) ≤ s. Suppose x ∈ / H and y ∈ / H. We consider the cases xy ∈ / H and xy ∈ H. Suppose xy ∈ / H. Then µ(xy) ∨ s = s ≥ µ(x) ∧ µ(y) ∧ t. Suppose / H. Thus xy ∈ H. Then µ(xy) ∨ s ≥ t ≥ µ(x) ∧ µ(y) ∧ t. Since x ∈ / H, x−1 ∈ −1 µ(x ) ∨ s = s ≥ µ(x) ≥ µ(x) ∧ t. Proposition 5.4.9. Let G be a group and µ an (s, t]-fuzzy subgroup of G. Then ∀x ∈ G, (µ(x) ∨ s) ∧ t = (µ(x−1 ) ∨ s) ∧ t. Proof. Let x ∈ G. Then µ(x−1 ) ∨ s ≥ µ(x) ∧ t and µ(x) ∨ s ≥ µ(x−1 ) ∧ t. Thus (µ(x−1 )∨s)∧t ≥ µ(x)∧t. Now (µ(x−1 )∨s)∧t = (µ(x−1 )∧t)∨(s∧t) = (µ(x−1 )∧ t) ∨ s. Hence (µ(x−1 ) ∧ t) ∨ s ≥ µ(x) ∧ t. Thus (µ(x−1 ) ∧ t) ∨ s ≥ (µ(x) ∧ t) ∨ s. Similarly, (µ(x)∧t)∨s ≥ (µ(x−1 )∧t)∨s. Thus (µ(x)∧t)∨s = (µ(x−1 )∨s)∧t. But (µ(x) ∧ t) ∨ s = (µ(x) ∨ s) ∧ (t ∨ s) = (µ(x) ∨ s) ∧ t. Hence (µ(x) ∨ s) ∧ t = (µ(x−1 ) ∨ s) ∧ t. Proposition 5.4.10. Let µ be an (s, t]-fuzzy subgroup of a group G. Then ∀x1 , ..., xn ∈ G, µ(x1 ...xn ) ∨ s ≥ µ(x1 ) ∧ ... ∧ µ(xn ) ∧ t. Proof. The proof is by induction on n. By Deﬁnition 5.4.1, the result is true for n = 2. Suppose the result is true for k ∈ N, k ≥ 2, the induction hypothesis. Let x1 , ..., xk+1 ∈ G. Then µ(x1 ...xk+1 ) ∨ s ≥ µ(x1 ...xk ) ∧ µ(xk+1 ) ∧ t. Thus µ(x1 ...xk+1 ) ∨ s ≥ (µ(x1 ...xk ) ∨ s) ∧ µ(xk+1 ) ∧ t ≥ (µ(x1 ) ∧ ... ∧ µ(xk ) ∧ t) ∧ µ(xk+1 ) ∧ t = µ(x1 ) ∧ ... ∧ µ(xk ) ∧ µ(xk+1 ) ∧ t. Proposition 5.4.11. Let {µi |i ∈ I} be a collection of (s, t]-fuzzy subgroups of a group G. Then ∩i∈I µi is an (s, t]-fuzzy subgroup of G. Proof. Let x, y ∈ G. Then (∩i∈I µi )(xy −1 ) ∨ s = ∧{µi (x) | i ∈ I} ∨ s = ∧{µi (xy −1 ) ∨ s | i ∈ I} ≥ ∧{µi (x)∧µi (y)∧t | i ∈ I} = (∧{µi (x) | i ∈ I})∧(∧{µi (y)∨s | i ∈ I}∧t) = (∩i∈I µi )(x) ∧ (∩i∈I µi )(y) ∧ t. Deﬁnition 5.4.12. Let η be a fuzzy subset of a group G. Let ≺ η ! denote the intersection of all (s, t]-fuzzy subgroups µ of G such that ∀x ∈ G, µ(x) ∨ s ≥ η(x) ∧ t. ∀x ∈ G, let λ(x) = η(x) ∨ η(x−1 ). ∀ z ∈ G, let rz = ∨{λ(z1 ) ∧ ... ∧ λ(zn ) | z = z1 ...zn , zi ∈ G, i = 1, ..., n; n ∈ N}.

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Lemma 5.4.13. Let η be a fuzzy subset of a group G. Let µ be an (s, t]-fuzzy subgroup of G such that ∀x ∈ G, µ(x) ∨ s ≥ η(x) ∧ t. Then the following assertions hold ∀x ∈ G : (1) µ(x) ∨ s ≥ λ(x) ∧ t; (2) If x = x1 ...xn for x1 , ..., xn ∈ G, then µ(x) ∨ s ≥ λ(x1 ) ∧ ... ∧ λ(xn ) ∧ t. Proof. (1) Since µ(x) ∨ s ≥ η(x) ∧ t and µ(x−1 ) ∨ s ≥ η(x−1 ) ∧ t, (µ(x) ∨ s) ∨ (µ(x−1 ) ∨ s) ≥ (η(x) ∧ t) ∨ (η(x−1 ) ∧ t) = λ(x) ∧ t. Hence ((µ(x) ∨ s) ∧ t) ∨ ((µ(x−1 ) ∨ s) ∧ t) ≥ λ(x) ∧ t. By Proposition 5.4.9, (µ(x) ∨ s) ∧ t ≥ λ(x) ∧ t. Thus µ(x) ∨ s ≥ λ(x) ∧ t. (2) By Proposition 5.4.10, µ(x)∨s ≥ µ(x1 )∧...∧µ(xn )∧t. Thus µ(x)∨s ≥ (µ(x1 ) ∨ s) ∧ ... ∧ (µ(xn ) ∨ s) ∧ t ≥ λ(x1 ) ∧ ... ∧ λ(xn ) ∧ t, where the latter inequality holds by (1). Theorem 5.4.14. Let η be a fuzzy subset of a group G. Let δ be the fuzzy subset of G deﬁned by ∀x ∈ G,  rx ≥ t,  t if δ(x) = rx if s < rx ≤ t,  0 if rx ≤ s, where rx is deﬁned immediately above. Then δ is a fuzzy subgroup of G and δ =≺ η ! . Proof. We ﬁrst show that δ is an fuzzy subgroup of G. Let x, y ∈ G. Then rxy = ∨{λ(z1 ) ∧ ... ∧ λ(zn )|xy = z1 ...zn ; zi ∈ G, i = 1, ..., n; n ∈ N} ≥ ∨{λ(x1 ) ∧ ... ∧ λ(xm ) ∧ λ(y1 ) ∧ ... ∧ λ(yk )|x = x1 ...xm , y = y1 ...yk , xi , yj ∈ G, i = 1, ..., m; j = 1, ..., k; m, n ∈ N} = rx ∧ ry ≥ δ(x) ∧ δ(y). Thus δ(xy) ≥ δ(x) ∧ δ(y) if s < rxy ≤ t. Suppose rxy ≥ t. Since δ(z) ≤ t∀z ∈ G, δ(xy) = t ≥ δ(x) ∧ δ(y). Suppose rxy ≤ s. Then rx ∧ ry ≤ s. Hence δ(xy) = 0. Now either rx ≤ s or ry ≤ s. Thus either δ(x) = 0 or δ(y) = 0. Hence δ(xy) ≥ δ(x) ∧ δ(y). For z ∈ G, z = z1 ...zn if and only if z −1 = zn−1 ...z1−1 . Thus it follows that rz = rz−1 . Let x ∈ G. Suppose s < rx−1 ≤ t. Then δ(x−1 ) = rx−1 = rx = δ(x). Suppose rx−1 ≥ t. Then rx ≥ t and so δ(x−1 ) = t = δ(x). Suppose rx−1 ≤ s. Then rx ≤ s and so δ(x−1 ) = 0 = δ(x). Hence δ is a fuzzy subgroup of G (and thus an (s, t]-fuzzy subgroup of G). Suppose η(x) ≥ t. Then rx ≥ t. Hence δ(x) = t = η(x) ∧ t. Thus δ(x) ∨ s ≥ η(x) ∧ t. Suppose s < η(x) ≤ t. Then either δ(x) = t or δ(x) = rx . Hence δ(x) ≥ rx ∧ t ≥ η(x) ∧ t. Thus δ(x) ∨ s ≥ η(x) ∧ t. Suppose η(x) ≤ s. Then δ(x) ∨ s ≥ s > η(x) ∧ t. Hence ∀x ∈ G, δ(x) ∨ s ≥ η(x) ∧ t.

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Now ≺ η ! is the smallest (s, t]-fuzzy subgroup of G such that ∀x ∈ G, ≺ η ! (x) ∨ s ≥ η(x) ∧ t. Hence ≺ η !⊆ δ. Let µ be any (s, t]-fuzzy subgroup of G such that ∀x ∈ G, µ(x)∨s ≥ η(x)∧t. In order to show, δ ⊆≺ η ! it suﬃces to show δ ⊆ µ. Let x ∈ G. By (2) of Lemma 5.4.13, it follows that µ(x) ∨ s ≥ rx ∧ t. Suppose rx ≥ t. Then µ(x) ∨ s ≥ rx ∧ t ≥ δ(x) ∧ t. Since s < t ≤ rx , µ(x) ≥ δ(x) ∧ t. But δ(x) ≤ t. Thus µ(x) ≥ δ(x). Suppose s < rx ≤ t. Then µ(x) ∨ s ≥ rx ∧ t = δ(x) ∧ t. Since s < rx , µ(x) ≥ δ(x) ∧ t. But δ(x) ≤ t. Thus µ(x) ≥ δ(x). Suppose rx ≤ s. Then δ(x) = 0 ≤ µ(x). Thus δ ⊆ µ. Hence δ ⊆≺ η ! . Thus δ =≺ η ! . Let G be a group and Y a subset of G. Recall that the notation Y is used to denote the subgroup of G generated by Y. Deﬁnition 5.4.15. Let η be a fuzzy subset of a group G. Deﬁne η to be the intersection of all (s, t]-fuzzy subgroups of G which contain η. If η is a fuzzy subset of a group G, then η is an (s, t]-fuzzy subgroup of G by Proposition 5.4.11. η is called the (s, t]-fuzzy subgroup of G generated by η. Let η be a fuzzy subset of a group G. Recall that the notation η is used to denote the intersection of all fuzzy subgroups of G which contain η. Proposition 5.4.16. Let η be a fuzzy subset of a group G. Then ≺ η !⊆ η ⊆ η. Proof. The proof follows from the deﬁnitions.

Deﬁnition 5.4.17. Let G and H be groups and let µ and ν be (s, t]-fuzzy subgroups of G and H, respectively. Let f be a homomorphism of G onto H. Then ν (or (H, ν)) is called an (s, t]-homomorphic image of µ (or (G, µ)) if ∀x ∈ G, f (µ)(f (x)) = ν(f (x)) if s < ν(f (x)) ≤ t, f (µ)(f (x)) ∧ t = ν(f (x)) ∧ t if ν(f (x)) > t, and f (µ)(f (x)) ∨ s = ν(f (x)) ∨ s if ν(f (x)) ≤ s. Let F be a group and µ a fuzzy subgroup of F. Let X be a subset of F and χ a fuzzy subset of X. Recall Deﬁnition 5.3.1; (F, µ) is called a free fuzzy group on (X, χ) provided the following conditions hold: (1) X generates F and χ generates µ; (2) Let G be a group and ν a fuzzy subgroup of G such that G = Y and ν = η. Then for any surjection f : X → Y such that f (χ) = η, there exists an epimorphism f ∗ : F → G such that f ∗ (x) = f (x) for all x ∈ X and f ∗ (µ) = ν. We point out that the fuzzy subset χ of X in Deﬁnition 5.3.1 can be considered to be a fuzzy subset of F by letting χ(x) = 0 for all x ∈ G, x ∈ / X. We make this type of assumption elsewhere without further comment. Let X be a set and χ a fuzzy subset of X. Then by Theorem 5.3.5, there exists a free fuzzy group on (X, χ) and if G is a group and ν a fuzzy subgroup of G, then (G, ν) is a homomorphic image of a free fuzzy group.

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Deﬁnition 5.4.18. Let F be a group and µ an (s, t]-fuzzy subgroup of F. Let X be a subset of F and χ a fuzzy subset of X. Then (F, µ) is called a free (s, t]-fuzzy group on (X, χ) provided the following conditions hold: (1) X generates F and χ generates µ, i. e., F = X and µ =≺ χ ! . (2) Let G be a group and ν an (s, t]-fuzzy group such that G = Y and ν =≺ η ! . Then for any surjection f : X → Y such that f (χ) = η, there exists an epimorphism f ∗ : F → G such that f ∗ (x) = f (x) for all x ∈ X and ν is an (s, t]-homomorphic image of µ under f ∗ . Theorem 5.4.19. Let X be a set and χ a fuzzy subset of X. Then there exists a free (s, t]-fuzzy group on (X, χ). Proof. Let (F, µ) be a free fuzzy group on (X, χ). Let (G, ν) be an (s, t]fuzzy group having a generating set (Y, η) and let f : (X, χ) → (Y, η) be a surjection. Let ν = η. Then there exists an epimorphism f ∗ : F → G such that f ∗ (x) = f (x) for all x ∈ X and f ∗ (µ) = ν . Let x ∈ G. Suppose that s < rf ∗ (x) ≤ t. Then f ∗ (µ)(f ∗ (x)) = ν (f ∗ (x)) = η(f ∗ (x)) = rf ∗ (x) =≺ η ! (f ∗ (x)) = ν(f ∗ (x)). Suppose rf ∗ (x) > t. Then η(f ∗ (x)) = rf ∗ (x) and ≺ η ! (f ∗ (x)) = t. Thus f ∗ (µ)(f ∗ (x)) = ν (f ∗ (x)) = η(f ∗ (x)) = rf ∗ (x) > t and ν(f ∗ (x)) =≺ η ! (f ∗ (x)) = t. Hence f ∗ (µ)(f ∗ (x)) ∧ t = ν(f ∗ (x)) ∧ t. Suppose that rf ∗ (x) ≤ s. Then f ∗ (µ)(f ∗ (x)) = ν (f ∗ (x)) = η(f ∗ (x)) = rf ∗ (x) ≤ s and ν(f ∗ (x)) =≺ η ! (f ∗ (x)) = 0. Thus f ∗ (µ)(f ∗ (x)) ∨ s = s = ν(f ∗ (x)) ∨ s. Hence (F, µ) is a free (s, t]-fuzzy group on (X, χ). Let µ0 be the fuzzy subgroup of G deﬁned by ∀x ∈ G, µ0 (x) = ν(x) if s < rx ≤ t, µ0 (x) = t if rx > t, and µ0 (x) = 0 if rx ≤ s. Let µ1 be an (s, t]-fuzzy subgroup of F such that ∀x ∈ G, µ1 (x) = µ0 (x) if s < rx ≤ t, µ0 (x) ≤ µ1 (x) ≤ ν(x) if rx > t or if rx ≤ s. Then (G, ν) is an (s, t]-fuzzy homomorphic image of (F, µ1 ). Thus we have the following result. Theorem 5.4.20. Let G be a group and ν a fuzzy subgroup of G. Then (G, ν) is an (s, t]-homomorphic image of a free (s, t]-fuzzy group. Remark 5.4.21. We feel that the notion of an (s, t]-fuzzy subgroup has possible applications for suitable s and t. For example, in medical diagnosis the presence or absence of a disease may be determined by certain high values, t, or certain low values, s, [22, 31]. The use of free algebraic structures in medical diagnosis can be found in [31]. An interesting problem would be to determine applications of (s, t]-fuzzy substructures of algebraic structures.

References 1. N. Ajmal, The free product is not associative, Fuzzy Sets and Systems 60 (1993) 241 - 244. 130 2. S. K. Bhakat, (∈ ∨q)-level subset, Fuzzy Sets and Systems 103 (1999) 529-533. 131

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6 Fuzzy Subgroups of Abelian Groups

Some of the best examples of algebraic structure theory come from commutative group theory. Commutative group theory is also a principal reason for the study of module theory. In this chapter, we present structure results for fuzzy subgroups of Abelian groups. Many of the results in this chapter have been applied to the study of fuzzy group subalgebras and fuzzy ﬁeld extensions [3, 11, 14, 15, 16, 21]. Throughout this chapter G is an (additive) Abelian group. Let H be a subgroup of G. Recall that H is called torsion if every element of H has ﬁnite order and H is called torsion-free if no element of H has ﬁnite order other than 0. The results in this chapter come mainly from [8, 9, 13, 16, 17]. For other approaches the reader is referred to [1, 5, 20].

6.1 Minimal Generating Sets and Direct Sums Let µ ∈ F(G). We introduce the concept of a minimal generating set for µ when the support of µ is a direct sum of cyclic groups. We show that if the support of µ is cyclic, then µ has a minimal generating set. However, we show that this result does not hold in general when the support of µ is a direct sum of two or more cyclic groups. Consequently, although a subgroup of a group G that is a direct sum of cyclic groups is a again a direct sum of cyclic groups, the corresponding result for fuzzy subgroups of G need not hold. Let I denote a nonempty index set. By the summation x = i∈I xi , we mean all but a ﬁnite number of the xi are zero. If {µ i | i ∈ I} is a collection of fuzzy subgroups of G, then deﬁne the fuzzy subset i∈I µi of G as follows: ∀x ∈ G, µi )(x) = ∨{∧{µi (xi )|i ∈ I}|x = xi }. ( i∈I

i∈I

This deﬁnition corresponds to the deﬁnition of weak product given in Deﬁnition 1.5.2. Thus the results in Chapter 1 concerning the weak product hold John Mordeson, Kiran R. Bhutani, Azriel Rosenfeld: Fuzzy Group Theory, StudFuzz 182, 139– 166 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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here with the appropriate changes in terminology and notation. We speak of · the weak direct sum rather than the weak direct product. We replace and ·

⊗ with ⊕. Deﬁnition 6.1.1. Let µ ∈ F(G) and let F(µ) = {ν ∈ F(G) | ν ⊆ µ}. Let χ be a fuzzy subset of G such that χ ⊆ µ. Deﬁne χ to be the intersection of all γ ∈ F(µ) such that χ ⊆ γ. Then χ is called a generating set for χ in µ. Clearly, χ is a fuzzy subgroup of G. In fact, χ is the smallest fuzzy subgroup of G in F(µ) containing χ. Deﬁnition 6.1.2. Let µ ∈ F(G). Let S denote a set of fuzzy singletons such that if xa , xb ∈ S, then a = b > 0 and x = 0. Deﬁne the fuzzy subset χ(S) of G as follows: ∀x ∈ G a if xa ∈ S χ(S)(x) = 0 otherwise. Set S = χ(S) . Then S is called a minimal generating set for µ if µ = S ∪ 0µ(0) and µ ⊃ S\{xa } ∪ 0µ(0) for all xa ∈ S. Theorem 6.1.3. Let µ ∈ F(G). If x1 , x2 , . . . , xn ∈ G are such that ∧n−1 i=1 µ(xi ) > µ(xn ), then µ(x1 + x2 + . . . + xn ) = µ(xn ). Proof. Let x = x1 + x2 + . . . + xn−1 . Then µ(x) ≥ ∧ni=1 µ(x) = ∧n−1 i=1 µ(xi ) > µ(xn ). Hence µ(xn ) = µ(−x + x + xn ) ≥ µ(x) ∧ µ(x + xn ) ≥ µ(x) ∧ µ(xn ) = µ(xn ). Thus µ(x) ∧ µ(x + xn ) = µ(xn ). Since µ(x) > µ(xn ), µ(x + xn ) = µ(xn ). Theorem 6.1.4. Let µ ∈ F(G) and µ(G)\{0} = {a0 , a1 , . . . , an }. Suppose that ai−1 > ai for all i ∈ I\{0}, where I = {0, 1, . . . , n}. If there exists a subgroup Hi of µai such that µai = µai−1 ⊕ Hi for all i ∈ I\{0}, then there exists µi ∈ L(G), i ∈ I, such that µ = ⊕i∈I µi , µ∗0 = µa0 , and µ∗i = Hi for all i ∈ I\{0}. Proof. Since µai = µai−1 ⊕ Hi for all i ∈ I\{0}, it follows that µan = µa0 ⊕ H1 ⊕ . . . ⊕ Hn . Deﬁne the fuzzy subset µi of G as follows: ∀x ∈ G, µi (0) = a0 , µi (x) = ai if x ∈ Hi \{0}, and µi (x) = 0 otherwise, i ∈ I, where H0 = µa0 . Then µi ∈ F(G) and µ∗i = Hi , i ∈ I. Clearly, µ0 = µ on H0 and µi = µ on Hi since Hi ⊆ µai and Hi ∩ µai−1 = {0}, i ∈ I\{0}. Let x ∈ µ∗ . Then x = x0 +x1 +. . .+xn , where xi ∈ Hi , i ∈ I. Hence µ(x) = µ0 (x)∧µ1 (x)∧. . .∧µn (x) by Theorem 6.1.3. Now

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( i∈I µi )(x) = ∨{∧ni=0 µ(yi )|x = y0 + y1 + . . . + yn , yi ∈ µ∗i , i ∈ I} = µ(x0 ) ∧ µ(x1 ) ∧ . . . ∧ µ(xn ) since the expression x = x0 + x1 + . . . + xn is unique. Hence µ = i∈I µi . Thus µ = ⊕i∈I µi by Theorem 1.5.9. Theorem 6.1.5. Let µ ∈ F(G) and µ(G)\{0} = {a0 , a1 , . . . , an }. Suppose that ai−1 > ai for all i ∈ I\{0}, where I = {0, 1, . . . , n}. If pµ∗ = {0} for p a prime, then µ is a weak direct sum of fuzzy subgroups whose supports are cyclic. Proof. Since pµ∗ = {0}, µai−1 is pure in µai and µai−1 is a direct summand of µai , say µai−1 ⊕ Hi for some subgroup Hi of G, i ∈ I\{0}. Thus by Theorem 6.1.4, µ∗ = µa0 ⊕ H1 ⊕ . . . ⊕ Hn and µ = ⊕i∈I µi for fuzzy subgroups µi of G such that µi ∗ = Hi , µ∗0 = µa0 = H0 . Since pµ∗ = {0}, each Hi is a weak direct sum of cyclic groups, say Hi = ⊕ji ∈Ji Kji , ji ∈ Ji (an index set), i = 0, 1, . . . , n. Deﬁne the fuzzy subsets ξji of G as follows: ∀x ∈ G, µi (x) if x ∈ Kji , ξji (x) = 0 otherwise, ji ∈ Ji , i = 0, 1, . . . , n. Then ξji is an fuzzy subgroup of G and ξj∗i = Kji is is the same constant ai on Kji \{0} and every µi equals cyclic. Since every ξj i ai on Hi \{0}, µi = ji ∈Ji ξji = ⊕ji ∈Ji ξji . Thus µ = ⊕i∈I (⊕ji ∈Ji ξji ). Proposition 6.1.6. Let µ ∈ F(G) and let χ and η be fuzzy subsets of G. If χ, η ⊆ µ, then χ + η ⊆ µ. Proof. Let z ∈ G. Then (χ+η)(z) = ∨{χ(x)∧η(y) | z = x+y} ≤ ∨{µ(x)∧µ(y) | z = x + y} ≤ ∨{µ(x + y) | z = x + y} = µ(z). Proposition 6.1.7. Let µ ∈ F(G). Let χ be an fuzzy subset of G. If χ ⊆ µ, then −χ ⊆ µ. Proof. By Deﬁnition 1.2.1, (−χ)(z) = χ(−z) ≤ µ(−z) = µ(z).

For any fuzzy singleton xa of G, it follows easily that −(xa ) = (−x)a . Theorem 6.1.8. Let µ ∈ F(G). Let χ be a fuzzy subset of G such that χ ⊆ µ. Let σ be the fuzzy subset of G deﬁned by ∀ y ∈ G, σ(y) = ∨{(e1 (x1 )a1 + . . . + en (xn )an )(y) | xi ∈ G, ai ∈ L, χ(xi ) = ai , ei = ± 1, i = 1, . . . , n; n ∈ N}. Then σ = χ . Proof. By Proposition 6.1.7, ei (xi )ai ⊆ χ for i = 1, . . . , n. Thus σ ⊆ χ since e1 (x1 )a1 + . . . + en (xn )an ⊆ χ by Proposition 6.1.6. To show that χ ⊆ σ, it suﬃces to show that σ ∈ F(µ) and χ ⊆ σ. Let x ∈ G and χ(x) = a. Then clearly xa (x) ≤ σ(x) since xa = e(xa ) with e = 1 and n = 1. Hence χ ⊆ σ. Let x, y ∈ G. Then σ(x) and σ(y) are supremums of numbers of

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the form (e1 (x1 )a1 + . . . + en (xn )an )(x) and (f1 (y1 )b1 + . . . + fm (ym )bm )(y), respectively. Suppose that σ(x) > 0 and σ(y) > 0. Then ∃ sequences aj = a1j ∧ . . . ∧ anj and bk = b1k ∧ . . . ∧ bmk such that aj → σ(x) and bk → σ(y). If x ∈ x1 , . . . , xn and y ∈ y1 , . . . , ym , then x + y ∈ x1 , . . . , xn , y1 , . . . , ym . Thus σ(x + y) ≥ ∨{aj ∧ bk | j, k = 1, 2, . . .} = (∨{aj | j = 1, 2, . . .}) ∧ (∨{bk | k = 1, 2, . . .}) = σ(x) ∧ σ(y). Clearly, σ(x) = σ(−x). Hence σ ∈ F(µ). Corollary 6.1.9. Let x ∈ G and a ∈ [0, 1]. Then ∀ y ∈ G, xa (y) = a if y / x . ∈ x and xa (y) = 0 if y ∈ Proof. xa (y) = ∨{(e1 (x)a + . . . + en (x)a )(y) | ei = ±1, i = 1, . . . , n; n ∈ N} = ∨{(m(x)a )(y) | m ∈ Z}. Corollary 6.1.10. Let µ ∈ F(G). Let S be a set of fuzzy singletons such that ∗ ∀xa ∈ S, 0 < a ≤ µ(x). Then S = S ∗ . ∗

Proof. x ∈ S ⇔ S (x) > 0 ⇔ ∨{(e1 (x1 )a1 +. . .+ek (xk )ak )(x) | (xi )ai ∈ S, ei = ±1, i = 1, . . . , k, k ∈ N} > 0 ⇔ x = e1 x1 +. . .+ek xk for some (xi )ai ∈ S, i = 1, . . . , k ⇔ x ∈ S ∗ . Recall that by a sum of the form u∈Im(µ) mu (nu x)au , where mu , nu ∈ N ∪ {0}, we mean only a ﬁnite number of mu are diﬀerent from 0. Lemma 6.1.11. Let µ ∈ F(G). Suppose that µ∗ = x for some x ∈ µ∗ . For all u ∈ µ(G)\{0}, let nu denote the smallest positive integer such that µu = nu x if µu = {0} and let au = µ(nu x). Then the following assertions hold: (1) u = au ; (2) µ = {(nu x)au |u ∈ µ(G)} . Proof. (1) au = µ(nu x) ≥ u. Let z ∈ µ∗ \{0} be such that µ(z) = u. Then z ∈ µu and so ∃m ∈ Z such that z = m(nu x). Thus u = µ(z) ≥ µ(nu x) = au . Hence u = au . ∗ (2) Clearly, {(n u x)au |u ∈ µ(G)} ⊆ µ. Let z ∈ µ . Now {(nu x)au |u ∈ µ(G)} (z) = ∨{( u∈µ(G) mu (nu x)au )(z) | mu ∈ Z}. Hence if µ(z) = u, then z ∈ µu . Thus z = m(nu x) for some m ∈ Z. Clearly, then µ(z) ≤ {(nu x)au |u ∈ µ(G)} (z). Let χ be a fuzzy subset of G such that χ ⊆ µ, where µ ∈ F(G). Let S(χ) denote the intersection of all fuzzy subgroups of G in F(µ) such that xa ⊆ µ ∀ xa ∈ S(χ). Then clearly S(χ) = χ . Theorem 6.1.12. Let µ ∈ F(G). Suppose that µ∗ = x for some x ∈ µ∗ . For all u ∈ µ(G), let nu denote the smallest positive integer such that µu = nu x . Then {(nu x)u | u ∈ µ(G)\{0}} is a minimal generating set for µ.

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Proof. Let I = Im(µ)\{0}. By Lemma 6.1.11, {(nu x)u | u ∈ I} is a set of generators of µ. Let v ∈ I and ν = {(nu x)u |u ∈ I}\{(nv x)v } . Then ν ∈ L(µ). Now v = µ(nv x) and ν(nv x) = ∨{u1 ∧ . . . ∧ un } | nv x = u∈I\{v} mu (nu x), mu ∈ Z, mu = 0 for u ∈ {u1 , . . . , un }, n ∈ N} = ∨{u| nv x = mu (nu x), u < v, u ∈ I} ( if ∃ u ∈ I such that u < v) < v, where = and < occur since I is well-ordered under < . Thus ν ⊂ µ. Now suppose there does knot exist u ∈ I such that u < v. Suppose that ν(nv x) > 0. Then nv x = i=1 mui (nui x) for some nui , mui , (ui > v), i = 1, . . . , k. Hence nv x ∈ nu x = µu for some u > v, a contradiction. Thus ν(nv x) = 0. Hence ν ⊂ µ. Proposition 6.1.13. Let µ ∈ F(G) and let {µj | j ∈ J} ⊆ F(µ), where J is a nonempty index set. Suppose that µ∗ = ⊕j∈J (µj )∗ , where ∀ j ∈ J, (µj )∗ = xj for some xj ∈ G. Let Sj = {((nj )uj xj )uj | uj ∈ µj (G), (nj )uj ∈ N} be a minimal generating set for µj , j ∈ J. Then the following assertions hold: (1) ∪j∈J Sj is a minimal generating set for ⊕j∈J µj ; (2) ∪j∈J Sj is a minimal generating set for µ if and only if µ = ⊕j∈J µj . ∈ J. Let z ∈ (µm )∗ for some Proof. Let ν = ⊕j∈J µj . Then ν(0) = µj (0) ∀j m ∈ J. Then ν(z) = ∨{∧{µj (zj )|j ∈ J} | z = j∈J zj , zj ∈ (µj )∗ ∀j ∈ J} = ∧{µm (z) ∧ µj (0) | j ∈ J\{m}} = µm (z) since the representation∗ z = j∈J zj is unique and ν(0) = µj (0) ∀j ∈ J. That is, ν = µm on (µm ) . Let yj ∈ Sm . Then ∪j∈J Sj > = ⊕j∈J µj and so ∪j∈J Sj \{yj } (y) = (⊕j∈J,m =j Sj ⊕ Sm \{yj })(y) = Sm \{yj } (y) < Sm (y) = ν(y). Thus ∪j∈J Sj is a minimal generating set for ν. Proposition 6.1.13 shows that if µ∗ is a direct sum of cyclic groups, then the question of when µ has a minimal set of generators is equivalent to that of when µ is a direct sum of fuzzy subgroups whose supports are cyclic. Example 6.1.14. Let G = Z(p)⊕Z(p2 ) and let H = (1, p) . Deﬁne the fuzzy subsets µ, ν, and γ of G by µ(x) = 1 if x ∈ H, µ(x) = 12 otherwise; ν((0, 0)) = 1, ν(x) = 12 if x ∈ Z(p) ⊕ {0}\{(0, 0)}, ν(x) = 0 otherwise; γ(0, 0) = 1, γ(x) = 12 if x ∈ {0} ⊕ Z(p2 )\{(0, 0)}, γ(x) = 0 otherwise. Then µ, ν, γ are fuzzy subgroups G such that ν, γ ⊆ µ and µ∗ = ν ∗ ⊕ γ ∗ . However, µ = ν + γ since µ((1, p)) = 1, while (ν +γ)((1, p)) = 12 . Now Im(ν) = {1, 12 , 0} = Im(γ), ν1 = {(0, 0)}, ν 21 = Z(p) ⊕ {0}, γ1 = {(0, 0)}, γ 12 = {0} ⊕ Z(p2 ). Thus {n 12 (1, 0) 12 } and {m 12 (0, 1) 12 } are minimal generating sets for ν and γ respectively, where n 12 = 1 = m 12 . Now {(1, 0) 12 } ∪ {(0, 1) 12 } is a minimal generating set for ν + γ, but not a generating set for µ since µ = ν + γ. However, µ = 1H ⊕ γ and so µ has a minimal generating set. Theorem 6.1.15. Suppose that G = ⊕m i=1 Gi , where Gi is a cyclic subgroup of G for i = 1, . . . , m. Suppose that ∃ a cyclic subgroup H of G such that H is not contained in any cyclic direct summand of G. Then ∃ an fuzzy subgroup µ of G such that µ is not a direct sum of fuzzy subgroups of G whose supports are cyclic.

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Proof. Deﬁne the fuzzy subset µ of G by µ(x) = 1 if x ∈ H and µ(x) = 12 otherwise. Then µ is a fuzzy subgroup of G and µ∗ = G. By hypothesis ∃ µi ∗ ∗ ∈ F(µ) for i = 1, . . . , m such that µ∗= ⊕m i=1 µi and µi = ci for some ci ∈ m G, i = 1, . . . , m. We show thatµ = i=1 µi so that µ = ⊕m i=1 µi . Let z ∈ G m be such that H = z .Then ( i=1 µi )(z) = ∨{∧{µi (zi )|i = 1, . . . , m} | z = m Thus ( i=1 µi )(z) = 0 if zi ∈ / µ∗i for some i, while µ(z) = 1. z1 + . . . +zm }. m Suppose that ( i=1 µi )(z) = 1. Then ∃ z1 , . . . , zm ∈ G such that z = z1 + . . . +zm and µi (zi ) = 1 for i = 1, . . . , m. Since µi ⊆ µ, zi ∈ H ∩ µ∗i , i = 1, . . . , m. Suppose that |H| = r. Then z has order r and so zi has order r for some i. Since zi ∈ H ∩ µ∗i , H = zi ⊆ µ∗i . Thus H is contained in a direct summand of G, a contradiction. Suppose that |H| = ∞. Then zi = qi z for some integers qi , i = 1, . . . , m. If qi = 0 = qj for i = j, then µ∗i ∩ µ∗j ⊇ qi z ∩ qj z ⊃ {0}, ∗ a contradiction. Hence ∃ i such i = ±1. Thus H ⊆ µi , mthat ∀ j, j = i, qj = 0,qm a contradiction. Therefore, ( i=1 µi )(z) = 1 and so i=1 µi ⊂ µ. Example 6.1.16. Let G = G1 ⊕ G2 , where Gi is a cyclic subgroup of G for i = 1, 2. Suppose that |G1 | = ∞ and |G2 | = pr for some prime p. Then we show that ∃ a fuzzy subgroup µ of G such that µ is not a direct sum of fuzzy subgroups of G whose supports are cyclic. Let Gi = xi , i = 1, 2. Let H = (px1 , x2 ) . Let K be a subgroup of G such that H ∩ K = {(0, 0)}. If (s1 x1 , s2 x2 ) ∈ K for s1 , s2 ∈ Z, then (pr+1 s1 x1 , 0) ∈ H ∩ K and so s1 = 0. Thus if H is contained in a direct summand of G, say G = H ⊕ K with H ⊆ H , then K ⊆ G2 . Hence K = G2 since G2 is a torsion subgroup of G. Also H is cyclic and of inﬁnite order. Let H = (t1 x1 , t2 x√2 ) . Then ∃ q ∈ Z such that (px1 , x2 ) = q(t1 x1 , t2 x2 ). Hence qt1 = p and p qt2 . Thus q = ±1 and so t1 = ∓ p. Hence H = H . But then (x1 , 0) ∈ / H ⊕ K, a contradiction. That is, H is not contained in a direct summand of G. Hence the desired result follow from Theorem 6.1.15. Example 6.1.17. Let G = G1 ⊕ G2 , where Gi is a cyclic subgroup of G, i = 1, 2 and |G1 | = pr , |G2 | = ps for s − r ≥ 2. Then we show that ∃ a fuzzy subgroup µ of G such that µ is not a direct sum of fuzzy subgroups of G whose supports are cyclic. Let Gi = xi , i = 1, 2. Let H = (x1 , pk x2 ) for 0 < k < s − r. Let K and J be any cyclic subgroups of G such that |K| = pr , |J| = ps , and G = K ⊕ J. Now J = (c, d) for some c ∈ G1 and d ∈ G2 . Now |H| = ps−k and s − k > r. Thus H ⊆ K. Suppose that H ⊆ J. Then ∃ q ∈ Z such that q(c, d) = (x1 , pk x2 ). Now o(qc) = pr . Thus pq. Hence o(qd) = o(d) = ps , a contradiction. Thus H is not contained in a direct summand of G. The result now follows from Theorem 6.1.15.

6.2 Independent Generators It is convenient to have available the concept of linear independence of a set of fuzzy singletons for the purpose of selecting a basis for a direct sum of cyclic fuzzy subgroups.

6.2 Independent Generators

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Let n ∈ N and xa be a fuzzy singleton in G. Then nxa denotes, x + . . . + xa = (nx)a . a n times

Deﬁnition 6.2.1. Let µ ∈ F(G). A system of fuzzy singletons {(x1 )a1 , . . . , (xk )ak }, where 0 < ai ≤ µ(xi ) for i = 1, . . . , k is said to be linearly independent in µ if n1 (x1 )a1 + . . . + nk (xk )ak = 0a implies n1 x1 = . . . = nk xk = 0 where ni ∈ Z for i = 1, . . . , k and a ∈ (0, 1]. A system of fuzzy singletons is called linearly dependent if it is not independent. An arbitrary system S of fuzzy singletons is independent in µ if every ﬁnite subsystem of S is independent. We let S denote a system of fuzzy singletons such that ∀ xa ∈ S, 0 < a ≤ µ(x). Let S ∗ = {x| xa ∈ S} and Sa = µa ∩ S ∗ ∀ a ∈ {a ∈ (0, 1] | a ≤ µ(0)}. Proposition 6.2.2. Let µ ∈ F(G). The following conditions are equivalent on S. (1) S is independent in µ; (2) S ∗ is independent in µ∗ ; (3) Sa is independent in µa for all a such that 0 < a ≤ µ(0). Proof. (1) ⇒ (2). Let n1 x1 + . . . + nk xk = 0, where x1 , . . . , xk ∈ S ∗ . Then n1 (x1 )a1 + . . . + nk (xk )ak = 0a , where (xi )ai ∈ S for i = 1, . . . , k and a = a1 ∧ . . . ∧ ak . Thus n1 x1 = . . . = nk xk = 0. (2) ⇒ (3). The result is immediate since Sa ⊆ S ∗ ∀ a ∈ {a ∈ (0, 1] | a ≤ µ(0)}. (3) ⇒ (1). Suppose that n1 (x1 )a1 + . . . + nk (xk )ak = 0a , where (xi )ai ∈ S for i = 1, . . . , k. Then a = a1 ∧ . . . ∧ ak , n1 x1 + . . . + nk xk = 0 and xi ∈ µa for i = 1, . . . , k. Thus n1 x1 = . . . = nk xk = 0. Deﬁnition 6.2.3. Let µ ∈ F(G). An independent system M of fuzzy singletons in µ is said to be maximal if an independent system S of fuzzy singletons in µ such that M ⊂ S. Proposition 6.2.4. Let µ ∈ F(G). Every independent system S in µ can be extended to a maximal independent system. Proof. By Proposition 6.2.2, S ∗ is independent in µ∗ . By Zorn’s Lemma, S ∗ can be extended to a maximal independent system M in µ∗ . Let M = {xa | x ∈ M, µ(x) = a}. Since x ∈ µ∗ , a > 0. Thus M∗ = M and so M is independent. Clearly, M is maximal since M∗ is maximal. Theorem 6.2.5. Let µ ∈ F(G). S is independent in µ if and only if the fuzzy subgroup of G generated by S in µ is a direct sum of fuzzy subgroups of G whose supports are cyclic, i.e., for S = {(xi )ai |0 < ai ≤ µ(xi ), i ∈ I}, S = ⊕i∈I (xi )ai . Proof. S = i∈I (xi )ai . Now S is independent in µ ⇔ S ∗ is independent ∗ in µ∗ ⇔ S ∗ = ⊕i∈I xi ⇔ S = ⊕i∈I (xi )ai since (xi )ai = xi ∀ i ∈ I.

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6.3 Primary Fuzzy Subgroups Deﬁnition 6.3.1. Let µ ∈ F(G). Then µ is called a p-primary fuzzy subgroup of G if ∃ a prime p such that ∀ fuzzy singletons xa ⊆ µ with a > 0, ∃ n ∈ N such that pn (xa ) = 0a . Let µ ∈ F(G). Then µ is p-primary if and only if µ∗ is p-primary. Also µ is p-primary if and only if µa is p-primary ∀a ∈ {a ∈ (0, 1] | a ≤ µ(0)}. For H a subgroup of G and p prime, let Hp denote the p-primary component of H. Proposition 6.3.2. Let µ ∈ F(G). Let p be a prime. Deﬁne the fuzzy subset µ(p) of G by ∀ x ∈ G, µ(p) (x) = µ(x) if x ∈ (µ∗ )p and µ(p) (x) = 0 otherwise. Then µ(p) is a p-primary fuzzy subgroup of G and (µ(p) )∗ = (µ∗ )p . Further more, (µ(p) )a = (µa )p ∀ a ∈ [0, 1], a ≤ µ(0). Proposition 6.3.3. Let µ ∈ F(G). Let p be a prime. Then µ(p) is the unique maximal p-primary fuzzy subgroup of G in µ. For p a prime, we call µ(p) the p-primary component of µ. Lemma 6.3.4. Let µ ∈ F(G). Let µi ∈ F(µ), i ∈ I. Suppose that µ∗ = ⊕i∈I µ∗i and that µ∗ is torsion. Suppose that ∀ x ∈ µ∗i , µ(x) = µi (x) ∀ i ∈ I. If the order of the elements of µ∗i are relatively prime to those of µ∗j (i = j, i, j ∈ I), then µ = ⊕i∈I µi . Proof. Let x ∈ µ∗ . Then x = i∈I xi , where xi ∈ µi ∗ and all but a ﬁnite number of xi = 0. By the assumption concerning the orders of the elements of µi ∗ , ∀ i ∈ I, ∃ ki ∈ N such that ki x = ki xi and ki , o(xi ) are relatively prime. µ(ki x) ≥ µ(x) ∀ i ∈ I. Thus Hence xi = ki xi and so µ(xi ) = µ(ki x i ) = µ )(x) = ∨{∧{µ (x ) | x = x , x ∈ µ∗i } | i ∈ I} = ∧{µi (xi ) | x ( i i i i∈I i i∈I i ) = ∧{µ(xi ) | x = i∈I xi } ≥ µ(x). Clearly, = i∈I xi } (since µ∗ = ⊕i∈I µ∗i ( i∈I µi )(x) ≤ µ(x). Thus µ = i∈I µi . Hence µ = ⊕i∈I µi by Theorem 1.5.9. Deﬁnition 6.3.5. Let µ ∈ F(G). µ is called a torsion fuzzy subgroup of G if ∀ fuzzy singletons xa ⊆ µ with a > 0, ∃n ∈ N such that n(xa ) = 0a . Proposition 6.3.6. Let µ ∈ F(G). The following assertions hold. (1) µ is torsion if and only if µ∗ is a subgroup of the torsion subgroup of G. (2) µ is torsion if and only if µa is torsion for all a such that 0 < a ≤ µ(0). (3) There exists a unique maximal fuzzy subgroup τ of G such that τ ⊆ µ and τ is torsion. Theorem 6.3.7. Let µ be a torsion fuzzy subgroup of G. Then µ is a direct sum of primary fuzzy subgroups of G.

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Proof. Since µ∗ is torsion, µ∗ = ⊕p (µ∗ )p , where (µ∗ )p is the p-primary component of µ∗ . By Propositions 6.3.2 and 6.3.3, µ(p) is the p-primary component of µ and (µ(p) )∗ = (µ∗ )p . Thus µ∗ = ⊕p (µ(p) )∗ . By Lemma 6.3.4, µ = ⊕p µ(p) . Theorem 6.3.8. Let µ, ν ∈ F(G) such that µ ⊆ ν. Then there is a unique maximal fuzzy subgroup γ of G such that µ ⊆ γ ⊆ ν and ∀xa ⊆ γ with a > 0, there exists n ∈ N and b ∈ (0, 1] such that nxb ⊆ µ. Proof. There exists a unique maximal subgroup T of ν ∗ such that T ⊇ µ∗ and T /µ∗ is torsion. Deﬁne the fuzzy subset γ of G by ∀x ∈ G, γ(x) = ν(x) if x ∈ T and γ(x) = 0 if x ∈ / T. Then γ is a fuzzy subgroup of G and µ ⊆ γ ⊆ ν. Let xa ⊆ γ with a > 0. Then x ∈ T and so there exists n ∈ N such that nx ∈ µ∗ . Thus nxb ⊆ µ for b = µ(nx) > 0. Let δ be any fuzzy subgroup of G such that µ ⊆ δ ⊆ ν. If xa ⊆ δ with a > 0 and nxb ⊆ µ for some n and b > 0, then δ(x) ≤ ν(x) = γ(x). Hence γ is the desired unique maximal fuzzy subgroup of G. The fuzzy subgroup γ of G in Theorem 6.3.8 is called the torsion closure of µ in ν. Theorem 6.3.9. Let µ, ν ∈ F(G) be such that µ ⊆ ν. Let γ be an fuzzy subgroup of G such that µ ⊆ γ ⊆ ν. Then γ is the torsion closure of µ in ν if and only if γ ∗ /µ∗ is the torsion subgroup of ν ∗ /µ∗ and γ = ν on γ ∗ . Proof. Suppose that γ is the torsion closure on µ in ν. Let x ∈ γ ∗ . Then xa ⊆ γ for some a > 0. Hence ∃n and b ∈ (0, 1] such that nxb ⊆ µ. Then nx ∈ µ∗ . Thus γ ∗ /µ∗ is torsion. Let x ∈ ν ∗ be such that ∃n such that nx ∈ µ∗ . Then xa ⊆ ν with a > 0 and nxb ⊆ µ for some b ∈ (0, 1]. Thus xa ⊆ γ and so x ∈ γ ∗ . Hence γ ∗ /µ∗ is the torsion subgroup of ν ∗ /µ∗ . Clearly, γ = ν on γ ∗ . Conversely, suppose that γ ∗ /µ∗ is the torsion subgroup of ν ∗ /µ∗ and γ = ν on γ ∗ . Let xa ⊆ γ with a > 0. Then x ∈ γ ∗ . Thus ∃n ∈ N and b ∈ (0, 1] such that nxb ⊆ µ. Now suppose that xa ∈ ν with a > 0 and nb x ⊆ µ for some n and b. Then x ∈ γ ∗ . Hence γ(x) = ν(x) ≥ a > 0. Thus xa ⊆ γ. Hence γ is the torsion closure of µ in ν.

6.4 Divisible and Pure Fuzzy Subgroups One of the most important types of subgroup of an Abelian group is a divisible group. The structure of a divisible group is completely known. A divisible subgroup of an Abelian group is a direct summand of that group. The notion of a pure subgroup is very useful in Abelian group theory. In this section, we study the notion of divisible and pure subgroups in a fuzzy setting. Deﬁnition 6.4.1. Let µ ∈ F(G). Then µ is called divisible if ∀xa ⊆ µ with a > 0, and ∀n ∈ N, ∃ya such that ya ⊆ µ such that n(ya ) = xa .

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Proposition 6.4.2. Let µ ∈ F(G). Then µ is divisible if and only if µa is divisible ∀a ∈ {a|0 < a ≤ µ(0)}. Proof. Suppose µ is divisible. Let a ∈ (0, µ(0)] and n ∈ N. Let x ∈ µa . Then xa ⊆ µ. Thus ∃ya ∈ µ such that n(ya ) = xa . Hence ny = x. Since µ(x) ≥ a, x ∈ µa . Thus µa is divisible. Conversely, suppose µa is divisible ∀a ∈ (0, µ(0)]. Let xa ⊆ µ with a > 0 and n ∈ N. Since x ∈ µa , ∃ y ∈ µa such that ny = x. Hence (ny)a = xa or n(ya ) = xa . Clearly ya ⊆ µ and so µ is divisible. Proposition 6.4.3. Let µ ∈ F(G). If µ is divisible, then µ∗ is divisible. Proof. Let x ∈ µ∗ and n ∈ N. Then µ(x) = a > 0 for some a ∈ (0, 1]. Thus xa ⊆ µ. Since µ is divisible, ∃ya ⊆ µ such that n(ya ) = xa . Hence ny = x. Since µ(y) ≥ a > 0, y ∈ µ∗ . Thus µ∗ is divisible. Proposition 6.4.4. Let µ ∈ F(G). If µ∗ is divisible and µ is constant on µ∗ \{0}, then µ is divisible. Proof. Let xa ⊆ µ with a > 0 and n ∈ N. Then x ∈ µ∗ and so ∃ y ∈ µ∗ such that x = ny. If y = 0, then x = 0 and the result follows. Let y = 0. Since µ is constant on µ∗ \{0}, µ(y) = µ(x) ≥ a. Thus ya ⊆ µ and xa = n(ya ). Hence µ is divisible. Let n ∈ N and xa ⊆ 1{0} . Then x = 0. Hence ∃ ya ⊆ 1{0} such that n(ya ) = xa . Thus 1{0} is divisible. Let T denote the torsion subgroup of G. Proposition 6.4.5. Let µ ∈ F(G). ∀x, y ∈ G and n ∈ N, ny = x implies that µ(x) = µ(y) for all divisible fuzzy subgroups µ of G if and only if G is torsion-free. Proof. Suppose that the condition concerning µ holds. Let x ∈ T. Then ∃n ∈ N such that nx = 0. Since 1{0} is divisible, 1{0} (x) = 1{0} (0) = 1. Hence x = 0. Thus T = {0}. Conversely, suppose that G is torsion-free. Let µ be any divisible fuzzy subgroup of G. Suppose that nx = y. Let a = µ(x). Then n(ya ) = xa . Now ∃y ∈ G such that ny a = xa and y ⊆ µ. Thus ny = x. Since G is torsion-free y = y and so ya ⊆ µ. Thus a = µ(x) = µ(ny) ≥ µ(y) ≥ a. Hence µ(x) = µ(y). Theorem 6.4.6. Let µ ∈ F(G). Let G = Q, where Q denotes the additive group of rational numbers. Then µ is divisible if and only if µ is constant on G. Proof. Suppose that µ is divisible. Let x ∈ G. Then ∃n ∈ N such that nx = m ∈ Z. By Proposition 6.4.5, µ(x) = µ(m). Now m·1 = m and so µ(m) = µ(1) by Proposition 6.4.5. Thus µ(x) = µ(1). Conversely, suppose that µ is constant on G. Let xa ⊆ µ and n ∈ N. Now ∃y ∈ G such that ny = x and so n(ya ) = xa . Since µ is constant on G, µ(y) = µ(x). Hence ya ⊆ µ. Thus µ is divisible.

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Proposition 6.4.7. Let G = Z(p∞ ). Then for all divisible fuzzy subgroups µ of G, ∀x, y ∈ G\{0}, ∀n ∈ N, ny = x implies µ(x) = µ(y). Proof. Let x, y ∈ G\{0}, n ∈ N and ny = x. If µ(x) = 0, then µ(x) = µ(y). Suppose that µ(x) = a > 0. Then xa ⊆ µ. Now ∃ya ⊆ µ such that n(ya ) = xa . Thus ny = x. Hence a = µ(x) = µ(ny) ≥ µ(y) ≥ a. Thus µ(y) = µ(x). Hence it suﬃces to show that if nw = x and nz = x, then µ(w) = µ(z) since we would have µ(y ) = µ(y) = µ(x) whenever ny = x. Let w = u/pr and z = v/ps , where u, p are relatively prime and v, p are relatively prime. Assume that r ≤ s. Suppose that n(u/pr ) = n(v/ps ) for some n ∈ N. If pr |n in Z, then n(u/pr ) = n(v/ps ) = 0 in Z(p∞ ) and so x = 0, a contradiction. Thus n = pt n , where 0 ≤ t < r and n and p are relatively prime. Hence n (u/pr−t ) = n (v/ps−t ). Thus 0 = n u = n (v/ps−r ) in Z(p∞ ). If r < s, then p|n v in Z, which is impossible. Hence r = s. Now ∃a, b ∈ Z such that 1 = au + bpr and so 1/pr = a(u/pr ) + b = au/pr in Z(p∞ ). Thus v/ps = v/pr = va(u/pr ). Hence µ(v/ps ) = µ(va(u/pr )) ≥ µ(u/pr ). By symmetry, µ(u/pr ) ≥ µ(v/ps ). We note that if G = Z(p∞ ), then µ(x) = µ(y) for all fuzzy subgroups µ of G and ∀x, y ∈ G\{0}. Theorem 6.4.8. Let µ ∈ F(G), where G = Z(p∞ ). Then µ is divisible if and only if µ is constant on G\{0}. Proof. Suppose that µ is divisible. The subgroups of G are chained, say {0} ⊂ H1 ⊂ . . . ⊂ Hi ⊂ . . . , where Hi is a cyclic group of order pi . Let x ∈ G\{0}. Then x has order pi for some i ∈ N. Let y = pi−1 x. By Proposition 6.4.7, µ(x) = µ(y). Now y has order p and H1 = y . By comments preceding the theorem, µ is constant, say a, on H1 \{0}. Hence µ(x) = a. Conversely, suppose that µ is constant on G\{0}. Let xa ⊆ µ with a > 0 and let n ∈ N. There exists y ∈ G such that ny = x. Hence nya = xa . However, µ(y) = µ(x) ≥ a. Thus ya ⊆ µ. Hence µ is divisible. Let

F (µ) = F(µ)\{1{0} }.

Let n ∈ N and let µ be a fuzzy subgroup of G. Deﬁne the fuzzy subset nµ of G by ∀x ∈ G, nµ(x) = ∨{µ(y)|x = ny, y ∈ G}. Then nµ is a fuzzy subgroup of G. In [20], µ is called divisible if µ = nµ∀n ∈ N. Suppose 0 < a ≤ µ(0). Then x ∈ (nµ)a ⇔ (nµ)(x) ≥ a ⇔ ∨{µ(y)|x = ny, y ∈ G} ≥ a ⇔ ∃y ∈ G such that x = ny and µ(y) ≥ a (assuming µ has the sup property) ⇔ x ∈ nµa . Thus if µ has the sup property, (nµ)a = nµa . Hence if µ has the sup property the two notions of divisibilty are equivalent as can be seen as follows. If µ has the sup property, µ is divisible with respect to Deﬁnition 6.4.1⇔ µa is divisible (Proposition 6.4.2) ∀a ∈ [0, µ(0)] ⇔ nµa = µa ∀a ∈ [0, µ(0)] ⇔ (nµ)a = µa ∀a ∈ [0, µ(0)] ⇔ µ = nµ. Deﬁnition 6.4.9. Let µ ∈ F(G). Then µ is called reduced if ν ∈ F (µ) such that ν is divisible.

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Proposition 6.4.10. Let µ ∈ F(G). Then µ is reduced if and only if there exists a such that 0 < a ≤ µ(0) and µa is reduced. Proof. Suppose that µ is not reduced. Then ∃ ν ∈ F (µ) such that ν is divisible. If ν(0) = µ(0), then deﬁne the fuzzy subset ν of G by ν (0) = µ(0) and ν (x) = ν(x) ∀x ∈ G\{0}. Then ν ∈ F (µ) and ν is divisible. Hence we may assume that ν(0) = µ(0). By Proposition 6.4.2, νa is divisible for all a such that 0 < a ≤ µ(0). Since νa ⊆ µa , µa is not reduced for all a such that 0 < a < µ(0). Conversely, suppose that for all a such that 0 < a ≤ µ(0), µa is not reduced. Then ∃ a divisible subgroup D(a) of µa for all a such that 0 < a ≤ µ(0). Let D = D(a) for a = µ(0). Then D ⊆ µa for all a such that 0 < a ≤ µ(0). Since D ⊆ µ∗ , µ(x) = µ(0) ∀x ∈ D. Deﬁne the fuzzy subset ν of G by ∀x ∈ D, ν(x) = µ(0) and ν(x) = 0 otherwise. Then ν is an fuzzy subgroup of G such that ν ⊆ µ. Now ∀xa ⊆ ν with a > 0 and ∀n ∈ N, ∃y ∈ D such that ny = x. Since ν(y) = µ(0) ≥ a, ya ⊆ ν. Thus ν is divisible. Hence µ is not reduced. Corollary 6.4.11. Let µ ∈ F(G). µ is reduced if and only if µ∗ is reduced. Proof. Suppose µ is reduced. Then µa is reduced for some a such that 0 < a ≤ µ(0). Now µ∗ ⊆ µa and so µ∗ is reduced. Conversely, suppose that µ∗ is reduced. Let a = µ(0). Deﬁnition 6.4.12. Let µ ∈ F(G). Let ν ∈ F(µ). Then ν is said to be pure in µ if ∀xa ⊆ ν with a > 0, ∀n ∈ N, ∀ya ⊆ µ, n(ya ) = xa implies ∃ za ⊆ ν such that n(za ) = xa . Proposition 6.4.13. Let µ ∈ F(G). Let ν ∈ F(µ). Then ν is pure in µ if and only if νa is pure in µa for all a such that 0 < a ≤ ν(0). Deﬁnition 6.4.14. Let χ be an fuzzy subset of G and let n ∈ N. Deﬁne the fuzzy subset nχ of G by ∀x ∈ G, (nχ)(x) = 0 if x ∈ / nG and (nχ)(x) = ∨{χ(y)|y ∈ G, x = ny}. Proposition 6.4.15. Let µ ∈ F(G). Let n ∈ N. Then (1) nµ(0) = µ(0); (2) nµ ⊆ µ; (3) nµ is a fuzzy subgroup of G; (4) If µ has the sup property, then nµ(G) ⊆ µ(G). Proof. (1) (nµ)(0) = ∨{µ(y)|0 = ny} = µ(0). (2) If x ∈ / nG, then (nµ)(x) = 0 ≤ µ(x). Suppose that x ∈ nG and x = ny for some y ∈ G. Then µ(x) = µ(ny) ≥ µ(y). Hence (nµ)(x) = ∨{µ(y)|y ∈ G, x = ny} ≤ µ(x). (3) Let x, y ∈ G. If either x ∈ / nG or y ∈ / nG, then (nµ)(x) ∧ (nµ)(y) = 0 ≤ (nµ)(x − y). Suppose that x ∈ nG and y ∈ nG. Then x − y ∈ nG and (nµ)(x − y) = ∨{µ(w)|w ∈ G, x − y = nw} ≥ ∨{µ(u − v)|u, v ∈ G, x = nu, y =

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nv} ≥ ∨{µ(u)∧µ(v)|x = nu, y = nv} = (∨{µ(u)|x = nu}) ∧(∨{µ(v)|y = nv}) = (nµ)(x) ∧ (nµ)(y). Thus nµ is an fuzzy subgroup of G. (4) Let a ∈ (nµ)(G). Then (nµ)(x) = a for some x ∈ G. Now (nµ)(x) = ∨{µ(y)|x = ny, y ∈ G} = a. Since µ has the sup property, ∃y ∈ G such that µ(y) = a. Hence a ∈ µ(G). Thus (nµ)(G) ⊆ µ(G). Suppose that µ∗ is torsion-free. If x ∈ nµ∗ , n ∈ N, then (nµ)(x) = µ(w) for some unique w ∈ µ∗ such that x = nw. Proposition 6.4.16. Let µ ∈ F(G) and let n ∈ N. Then the following assertions hold. (1) (nµ)∗ = nµ∗ ; (2) nµa ⊆ (nµ)a ∀a such that 0 < a ≤ µ(0). If either µ has the sup property or µ∗ is torsion free, then nµa = (nµ)a ∀a such that 0 < a ≤ µ(0). (3) Let ν ∈ F(µ) and ν(0) = µ(0). If µ has the sup property and ν is pure in µ, then nνa = (nν)a for all a such that 0 < a ≤ µ(0). Proof. (1) x ∈ (nµ)∗ ⇔ (nµ)(x) > 0 ⇔ x = nw for some w ∈ µ∗ ⇔ x ∈ nµ∗ . (2) Let x ∈ nµa . Then x = nw for some w ∈ µa . Hence (nµ)(x) ≥ a. Thus x ∈ (nµ)a . Conversely, let x ∈ (nµ)a . Now (nµ)(x) = ∨{µ(y)|x = ny, y ∈ G}. Thus ∃y ∈ G such that x = ny and µ(y) ≥ a. Hence x ∈ nµa . (3) Let a ∈ {a|0 < a ≤ µ(0)}. Since ν is pure in µ, νa is pure in µa . Thus nνa = νa ∩ nµa . Let x ∈ (nν)a . Then (ν ∩ nµ)(x) ≥ (nν)(x) ≥ a. Thus x ∈ (ν ∩ nµ)a = νa ∩ (nµ)a = νa ∩ nµa = nνa . Hence (nν)a ⊆ nνa . Proposition 6.4.17. Let µ ∈ F(G) and let ν ∈ F(µ) be such that ν(0) = µ(0). Then the following assertions. (1) Suppose that µ has the sup property. If ν is pure in µ, then ∀n ∈ N, nν = ν ∩ nµ. (2) Suppose that µ has the sup property. If ∀n ∈ N, nν = ν ∩ nµ, then ν is pure in µ. (3) Suppose that µ∗ is torsion-free. Then ν is pure in µ if and only if ∀n ∈ N, nν = ν ∩ nµ. Proof. ν is pure in µ ⇔ ∀a such that 0 < a ≤ µ(0), νa is pure in µa ⇔ ∀a such that 0 < a ≤ µ(0), ∀n ∈ N, nνa = νa ∩ nµa ⇔ ∀a such that 0 < a ≤ µ(0), ∀n ∈ N, (nν)a = νa ∩ (nµ)a ⇔ ∀a such that 0 < a ≤ µ(0), ∀n ∈ N, nνa = (ν ∩ nµ)a ⇔ ∀n ∈ N, nν = ν ∩ nµ. Let ν be a fuzzy subgroup of G. In [20], ν is said to to be pure in G if nν = ν ∩ n(1G )∀n ∈ N. It follows easily that n(1G ) = 1nG . Hence it follows from Propositions 6.4.13, 6.4.16 and 6.4.17 that the notions of purity in G are equivalent if µ has the sup property. The notion of p-pure fuzzy subgroups is also studied in [20]. Proposition 6.4.18. Let µ ∈ F(G) and let ν, γ ∈ F(µ) be such that γ ⊆ ν and γ(0) = ν(0) = µ(0). Then the following assertions hold.

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(1) If γ is pure in ν and ν is pure in µ, then γ is pure in µ. (2) If ν is divisible, then ν is pure in µ. (3) Suppose that µ is divisible. Then ν is pure in µ if and only if ν is divisible. Proof. (1) Suppose that nya = xa , where ya ⊆ µ, xa ⊆ γ, and n ∈ N. Since xa ⊆ ν, ∃za ⊆ ν such that nza = xa . Hence ∃ wa ⊆ γ such that nwa = xa . (2) νa is divisible ∀a such that 0 < a ≤ ν(0). Thus νa is pure in µa ∀a such that 0 < a ≤ µ(0). Hence ν is pure in µ. (3) µa is divisible ∀a such that 0 < a ≤ µ(0). Thus νa is pure in µa ∀a such that 0 < a ≤ µ(0) ⇔ νa is divisible ∀a such that 0 < a ≤ µ(0). That is, ν is pure in µ ⇔ ν is divisible. Proposition 6.4.19. Let µ ∈ F(G). Let ν ∈ F(µ) be such that ν(0) = µ(0). If ∀xa ⊆ µ such that xa ⊆ν, ∃ n ∈ N such that n(xa ) ⊆ ν, then ν is pure in µ. Proof. If ∃a such that 0 < a ≤ µ(0), ∃ x ∈ µa such that x ∈ / νa and nx = z ∈ νa for some z ∈ νa and for some n ∈ N, then we have a contradiction of the hypothesis. Thus µa /νa is torsion-free and so νa is pure in µa ∀ a such that 0 < a ≤ µ(0), ([4], p. 114). Thus ν is pure in µ. Proposition 6.4.20. Let µ ∈ F(G). Suppose that µ∗ is torsion-free. Let {νj | j ∈ J} be a collection of fuzzy subgroups of G such that νj ⊆ µ, νj is pure in µ, and νj (0) = µ(0) ∀j ∈ J. Then ∩j∈J νj is pure in µ. Proof. Let x ∈ µ∗ . Then (∩j∈J nνj )(x) = ∧{nνj (x)|j ∈ J} = 0 if x ∈ / nµ∗ and (∩j∈J nνj )(x) = ∧{nνj (x)|j ∈ J} = ∧{νj (y)|j ∈ J} if x = ny ∈ nµ∗ . / nµ∗ and (n ∩j∈J νj )(x) = (∩j∈J νj )(y) = Now (n ∩j∈J νj )(x) = 0 if x ∈ ∗ ∧{νj (y)|j ∈ J} if x = ny ∈ nµ . Thus n ∩j∈J νj = ∩j∈J nνj . Now (∩j∈J (νj ∩ nµ))(x) = ∧{(νj ∩ nµ)(x) | j ∈ J} = ∧{νj (x) ∧ (nµ)(x) | j ∈ J} = ∧{νj (x) | j ∈ J} ∧ (nµ)(x) = (∩j∈J νj )(x) ∧ (nµ)(x) = ((∩j∈J νj ) ∩ (nµ))(x). Hence ∩j∈J (νj ∩ nµ) = (∩j∈J νj ) ∩ (nµ). ∀j ∈ J, ∀n ∈ N, nνj = νj ∩ nµ. Thus ∀n ∈ N, ∩j∈J nνj = ∩j∈J (νj ∩ nµ) and so n ∩j∈J νj = (∩j∈J νj ) ∩ nµ. Proposition 6.4.21. Let µ ∈ F(G). Suppose that µ∗ is torsion-free. Let ν ∈ F(µ) be such that ν(0) = µ(0). Then ν is contained in a unique smallest pure fuzzy subgroup in µ. Proof. Let B = {γ| ν ⊆ γ ⊆ µ, γ is a pure fuzzy subgroup of G in µ}. Then µ ∈ B and so B = ∅. The desired result now follows since ∩γ∈B γ is an fuzzy pure subgroup of G in µ by Proposition 6.4.20. Lemma 6.4.22. Let {νj |j ∈ J} and {γj |j ∈ J} be chains of fuzzy subgroups of G. Let n ∈ N. Then the following assertions hold. (1) n(∪j∈J νj ) = ∪j∈J nνj . (2) ∪j∈J (νj ∩ γj ) = (∪j∈J νj ) ∩ (∪j∈J γj ).

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Proof. (1) Let x ∈ G. Then n(∪j∈J νj )(x) = 0 if x ∈ / nG and n(∪j∈J νj )(x) = ∨{∪j∈J νj (y) | x = ny} = ∨{∨{νj (y) | j ∈ J} | x = ny} if x ∈ nG. Now / nG and (∪j∈J nνj )(x) = (∪j∈J nνj )(x) = ∨{(nνj )(x) | j ∈ J} = 0 if x ∈ ∨{(nνj )(x) | j ∈ J} = ∨{∨{νj (y) | x = ny} | j ∈ J} if x ∈ nG. Thus we have the desired result. (2) Let x ∈ G. Then (∪j∈J (νj ∩ γj ))(x) = ∨{(νj ∩ γj )(x) | j ∈ J} = ∨{νj (x) ∧ γj (x) | j ∈ J} = (∨{νj (x) | j ∈ J}) ∧ (∨{γj (x) | j ∈ J}) = (∪j∈J νj )(x) ∧ (∪j∈J γj )(x) = (( ∪j∈J νj )∩ (∪j∈J γj ))(x). Proposition 6.4.23. Let µ ∈ F(G). Let {µj |j ∈ J} be a chain of fuzzy subgroups of G such that µj ⊆ µ, µj is pure in µ, and µj (0) = µ(0) ∀j ∈ J. Suppose that either µ and ∪j∈J µj have the sup property or µ∗ is torsion-free. Then ∪j∈J µj is a pure fuzzy subgroup of G in µ. Proof. Let n ∈ N. Then n∪j∈J µj = ∪j∈J nµj = ∪j∈J (µj ∩nµ) = (∪j∈J µj )∩nµ by Lemma 6.4.22 and Proposition 6.4.17. Proposition 6.4.24. Let µ ∈ F(G). Let ν, γ, δ ∈ F(µ) be such that δ ⊆ ν and δ(0) = µ(0). Suppose that µ = ν ⊕ γ. Then the following assertions hold. (1) ∀n ∈ N, nµ = nν ⊕ nγ. (2) δ + γ = δ ⊕ γ and ν ∩ (δ ⊕ γ) = δ. Proof. (1) Let x ∈ µ∗ . Then (nµ)(x) = ∨{µ(y)|y ∈ G, x = ny} = ∨{(ν ⊕γ)(x) | x = ny} = ∨{ν(y1 ) ∧ γ(y2 ) | x = ny, y = y1 + y2 for y1 ∈ ν ∗ , y2 ∈ γ ∗ } since µ∗ = ν ∗ ⊕ γ ∗ by Theorem 1.5.9. Now (nν ⊕ nγ)(x) = (nν(x1 ) ∧ (nγ)(x2 ) such that x1 ∈ ν ∗ , x2 ∈ γ ∗ . Thus (nν ⊕ nγ)(x) = (∨{ν(y1 )|x1 = ny1 }) ∧ (∨{γ(y2 )|x2 = ny2 }). Hence nµ ⊆ nν ⊕ nγ. Clearly, nν ⊕ nγ ⊆ nµ. (2) For all x ∈ G, 0 ≤ (δ ∩ γ)(x) ≤ (ν ∩ γ)(x) = 0. Thus δ + γ = δ ⊕ γ. Now ∀x ∈ G, (ν ∩ (δ ⊕ γ))(x) = ν(x) ∧ (δ ⊕ γ)(x) = ν(x) ∧ δ(x1 ) ∧ γ(x2 ) if x = x1 + x2 for x1 ∈ δ ∗ , x2 ∈ γ ∗ and (ν ∩ (δ ⊕ γ))(x) = 0 otherwise. Consider the former case. Suppose that x ∈ / ν ∗ . Then (ν ∩ (δ ⊕ γ))(x) = 0 = δ(x). ∗ Suppose that x ∈ ν . Then x2 = 0 and so (ν ∩ (δ ⊕ γ))(x) = ν(x) ∧ δ(x) ∧ γ(0) = ν(x) ∧ δ(x) = δ(x). Proposition 6.4.25. Let µ ∈ F(G). Let ν and γ ∈ F(µ). If µ = ν ⊕ γ, then ν and γ are pure in µ. Proof. Suppose that nya = xa , where n ∈ N, ya ⊆ µ, and xa ⊆ ν. Since µ = ν ⊕ γ, µa = νa ⊕ γa . Thus νa ∩ nµa = nνa . Now nya = (ny)a . Hence ny = x, y ∈ µa , and x ∈ νa . Thus ∃z ∈ νa such that nz = x. Hence nza = xa . Thus ν is pure in µ. Example 6.4.26. Let G = Z(p∞ ) ⊕ Z(p∞ ). Deﬁne the subsetsµ, ν, and fuzzy 1 1 ⊕{0})\( pn−2 ⊕ γ of G by ∀x ∈ G, µ((0, 0)) = 1, µ(x) = 12 + n1 if x ∈ ( pn−1 {0}) for n = 2, 3, . . . , µ(x) = 12 if x ∈ / Z(p∞ ) ⊕ {0}; ν((0, 0)) = 1, ν(x) = 12 ∞ if x ∈ {0} ⊕ Z(p )\{(0, 0)}, ν(x) = 0 otherwise; γ((0, 0)) = 1, γ(x) = 12 if

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x ∈ ( p1 , p1 ), . . . , ( p1i , p1i ), . . . \{(0, 0)}, γ(x) = 0 otherwise. Then µ, ν, and γ

∗ ∗ ∞ are fuzzy subgroups of G such that ν, γ ⊆ µ and µ = G, ν = {0} ⊕ Z(p ), 1 1 1 1 ∗ ∗ ∗ ∗ ∗ ∗ γ = ( p , p ), . . . , ( pi , pi ), . . . . Since (ν + γ) = ν + γ = ν ⊕ γ , ν + γ = ν ⊕ γ by Theorem 1.5.9. Since ν ∗ and γ ∗ are divisible and ν, γ are constants on ν ∗ \{(0, 0)}, γ ∗ \{(0, 0)}, respectively, ν and γ are divisible by Proposition 6.4.4. Now (ν ⊕ γ)∗ = G which is divisible. Let (u, v) ∈ G\{(0, 0)}. Then (ν ⊕ γ)(u, v) = ν((0, v − u)) ∧ γ(u, u)) = 12 . That is, (ν ⊕ γ) is constant on (ν ⊕ γ)∗ \{(0, 0)}. Hence ν ⊕ γ is divisible. a nontrivial fuzzy subgroup δ of G such that µ = (ν ⊕ γ) ⊕ δ else (ν ⊕ γ)∗ ⊕ δ ∗ = µ∗ = (ν ⊕ γ)∗ and so δ ∗ = {(0, 0)}. Clearly, µ = ν + γ. That is, we have shown that a divisible fuzzy subgroup need not be a direct summand (of µ) even though the corresponding result holds for crisp Abelian groups.

6.5 Invariants of Fuzzy Subgroups The description of a class of groups by means of invariants is one of the major goals of group theory. If G is a ﬁnitely generated commutative group, then G is a direct sum of cyclic groups of order inﬁnity and powers of primes and thus has a complete system of invariants, [[4], p. 81]. In this section, we determine a complete system of invariants for those fuzzy subgroups µ of G which are direct sums of fuzzy subgroups whose supports are cyclic. Our invariants satisfy the criteria that they are easily described quantities and are uniquely determined by the fuzzy subgroup. Recall that for x ∈ G, o(x) denotes the order of x. Let µ denote an fuzzy subgroup of G. Let S(µ) = {xa | x ∈ µ∗ , µ(x) = a} and FS(µ) = {χ | χ is an fuzzy subset of G and χ ⊆ µ}. The following deﬁnition corresponds to Deﬁnition 6.1.2. Deﬁnition 6.5.1. Let µ ∈ F(G) and let χ ∈ FS(µ). Then χ (or S(χ)) is said to be a minimal generating set for µ if µ = χ ∪ 0µ(0) and ∀ xa ∈ S(χ), χ − xa ∪ 0µ(0) ⊂ χ , where (χ − xa )(y) = (χ)(y) if y = x and (χ − xa )(y) = 0 otherwise. Proposition 6.5.2. Let µ ∈ F(G). Suppose that µ∗ = x = y . For all u ∈ µ(G), let nu , mu be the smallest positive integers such that µu = nu x = mu y . Then nu = mu ∀ u ∈ µ(G). Proof. There exist q1 , q2 ∈ Z such that x = q1 y and y = q2 x. Suppose that µ∗ has inﬁnite order. Now x = q1 q2 x and so q1 = q2 = ±1. Thus nu = mu ∀ u ∈ µ(G). Suppose that µ∗ has ﬁnite order q. Then q, q1 are relatively prime else ◦(x) < ◦(y). Thus nu y = nu q1 y = nu x . Similarly, mu x = mu y . Hence nu = mu ∀ u ∈ µ(G).

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155

Let µ ∈ F(G) and x ∈ µ∗ . Let µx be the fuzzy subset of G such that µ (z) = µ(z) ∀ z ∈ x and µx (z) = 0 ∀ z ∈ G\ x . Then µx is an fuzzy subgroup of G. Let Ix = {nu | u ∈ µx (G)\{0}, nu the smallest positive integer such that (µx )u = nu x}. Suppose that µ∗ = ⊕j∈A xj . Let Iµ denote the ordered pair, (∪j∈A {(Ixj , o(xj ))}, µ(G)\{0}). In the remainder of this section, G denotes an Abelian group. x

Lemma 6.5.3. Let µ ∈ F(G) and ν ∈ F(G ). Suppose that µ∗ = x and ν ∗ = y . Then Iµ = Iν if and only if ∃ isomorphism f of µ∗ onto ν ∗ such that f (µ) = ν on ν ∗ . If Iµ = Iν , then every isomorphism f of µ∗ onto ν ∗ is such that f (µ) = ν on ν ∗ . Proof. Suppose that Iµ = Iν . Then clearly µ∗ ν ∗ for some isomorphism f and we may take f so that f (x) = y. Then ∀ u ∈ µ(G), f (nu x) = nu y. Since µ(G)\{0} = ν(G )\{0}, f (µu ) = νu ∀ u ∈ µ(G)\{0}. Thus since f is one-to-one, ∀ z ∈ µ∗ , f (µ)(f (z)) = ∨{µ(w) | f (w) = f (z)} = µ(z) = ν(f (z)), where the last equality holds since f (µu ) = νu ∀ u ∈ µ(G)\{0}. Hence f (µ) = ν on ν ∗ . Conversely, suppose ∃ isomorphism f of µ∗ onto ν ∗ such that f (µ) = ν. Then ∀ z ∈ µ∗ , ν(f (z)) = f (µ)(f (z)) = µ(z) as above. Hence µ(G)\{0} = Im(ν)\{0}. Clearly, o(x) = o(y). Now Ix = Iy since νu = f (µ)u = f (µu ) ∀ u ∈ µ(G)\{0}. Thus Iµ = Iν . For any isomorphism f of µ∗ onto ν ∗ , where Iµ = Iν , the proof that f (µ) = ν follows in a similar manner as above since Iν is independent of the generator of ν ∗ by Proposition 6.5.2. Let µ ∈ F(G) and ν ∈ F(G ). Suppose that µ∗ = ⊕j∈A xj and ν ∗ = ⊕j∈A yj . By Theorem 6.1.12, µxj = {(nuj xj )uj |uj ∈ µxj (G)\{0}} and

ν yj = {(mvj yj )vj |vj ∈ ν yj (G)\{0}}

∀ j ∈ A. We let µj = µxj ∀ j ∈ A. The next two results show that Iµ is a complete system of invariants for µ when µ is a direct sum of fuzzy subgroups whose supports are cyclic and where the xj below have orders ∞ or powers of primes. There is no loss in generality in using the same index set A for µ and ν. Theorem 6.5.4. Let µ ∈ F(G) and ν ∈ F(G ). Suppose that µ = ⊕j∈A µj and ν = ⊕j∈A ν j , where µ∗ = ⊕j∈A xj , ν ∗ = ⊕j∈A yj . If Iµ = Iν , then ∃ isomorphism f of µ∗ onto ν ∗ such that f (µ) = ν. Proof. By Lemma 6.5.3, ∃ isomorphism fj of xj onto yj (assuming the xj and yj are suitably arranged) such that fj (xj ) = yj and fj (µj ) = ν j∀ j ∈ A. ∗ of µ onto ν ∗ . Now f (µ)( j mj yj ) Let f = ⊕j∈A fj . Then f is an isomorphism = f (µ)( j mj f (xj )) = f (µ)(f j mj xj ) = µ( j mj xj ) since f is one-to-one. Also µ( j mj xj ) = (⊕j∈A µj )( j mj xj ) = ∧{µj (mj xj ) | j ∈ A} (since µ∗ is

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a direct sum) = ∧{wj | wj ∈µj (G)\{0}, j ∈ A}, where wj = µ(0) if mj = 0 and wj is such that mj xj ∈ nwj xj \ nuj xj ∀ uj ∈ µj (G) such that uj > nuj ∈ Iyj , mj yj ∈ nwj yj \ nuj yj wj , nwj , nuj ∈ Ixj , otherwise. Now nwj , ν( mj y j ) = ∧{wj | wj ∈ Im(ν j )\{0}, ∀ uj ∈ ν j (G) such that uj > wj and so j ∈ A} in a similar manner. Thus µ( j mj xj ) = ν( j mj yj ). Hence f (µ) = ν. Theorem 6.5.5. Let µ ∈ F(G) and ν ∈ F(G ). Suppose that f is an iso∗ ∗ = ν. morphism of µ∗ onto ν ∗ such that f (µ) If µ j= ⊕j∈A xj , then ν = j ⊕j∈A f (xj ) , Iµ = Iν , and f ( j∈A µ ) = j∈A ν . Proof. Clearly, ν ∗ = ⊕j∈A f (xj ) , f (µj ) = ν j , and o(xj )= o(f (xj )) ∀ j ∈ A.By Lemma 6.5.3, Ixj = If (xj ) ∀ j ∈ A. Now ν( j mj f (xj )) = f (µ)(f j mj xj ) = µ( j mj xj ). Thus µ(G)\{0} = ν(G)\{0}. Hence Iµ = Iν . Now, µj ))( mj f (xj )) = (f ( µj ))(f ( mj xj )) (f ( j

j

j

j

=( µj )( mj xj ) j

j

= ∧{µ (mj xj )|j ∈ A}(since µ∗ is a direct sum) = ∧{f (µj )(f (mj xj ))|j ∈ A} j

= ∧{ν j (mj f (xj ))|j ∈ A} ν j )( mj f (xj )). =( j

Thus f ( j µj ) = j ν j .

Deﬁnition 6.5.6. Let µ ∈ F(G) and γ ∈ F(µ) and xa ⊆ µ. Then the fuzzy subset xa + γ is called the left fuzzy coset of γ in µ with representative xa . Using the deﬁnition of the sum of two fuzzy subsets, it follows easily that ∀ z ∈ G, (xa + γ)(z) = a ∧ γ(z − x). It also follows that for all a in [0, µ(0)], (µ/γ)(a) = {xa + γ | xa ⊆ µ, x ∈ G} is a commutative group under +, where (xa + γ) + (ya + γ) = (x + y)a + γ. Theorem 6.5.7. Let µ ∈ F(G) and γ ∈ F(µ). Deﬁne µ/γ = { xa + γ | xa ⊆ µ, x ∈ G, a ∈ [0, 1]}. Then (µ/γ, +) is a commutative semi-group with identity. If γ(0) = µ(0), then µ/γ is completely regular [18] and µ/γ = ∪a∈M (µ/γ)(a) , where M = {a | 0 ≤ a ≤ µ(0)}. Theorem 6.5.8. Let µ ∈ F(G) and γ ∈ F(µ). Then (µ/γ)(a) µa /γa ∀ a such that 0 ≤ a ≤ µ(0).

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157

Let f be a homomorphism of G into G . Then ∀ xa ⊆ 1G , f (xa )(f (x)) = ∨{xa (y) | f (y) = f (x)} = ∨{xa (x)} = a = f (x)a (f (x)). If f (z) = f (x), then f (xa )(f (z)) = ∨{xa (y) | f (y) = f (z)} = 0 = f (x)a (f (z)) since y = x. If y ∈ G \f (G), then f (xa ) = 0 by deﬁnition. Thus f (xa ) = f (x)a . Let µ, ν ∈ F(G). Let γ ∈ F(µ), δ ∈ L(ν), and f be an isomorphism of γ ∗ onto δ ∗ such that f (γ) = δ. Then (f (xa + γ))(f (z)) = ∨{(xa + γ)(y) | f (y) = f (z)} = (xa + γ)(z) (since f is one-to-one) = a ∧ γ(−x + z) = a ∧ δ(f (−x + z)) = a ∧ δ(−f (x) + f (z)) = (f (x)a + δ)(f (z)). That is, (f (xa + γ))(f (z)) = (xa + γ)(z) = (f (x)a + δ)(f (z)). This discussion is relevant to (3) of the next result. Theorem 6.5.9. Let µ, ν ∈ F(G), γ ∈ F(µ), and δ ∈ F(ν). Suppose that µ∗ = γ ∗ = ⊕j∈A xj , ν ∗ = δ ∗ = ⊕j∈A yj , γ = ⊕j∈A µα , and δ = ⊕j∈A ν j . Suppose further that Iγ = Iδ . Then ∃ isomorphism f of γ ∗ onto δ ∗ such that f (xj ) = yj ∀ j ∈ A and f (γ) = δ. Also the following conditions are equivalent. (1) f (µ) = ν; (2) ∀ a such that 0 ≤ a ≤ µ(0), xa ⊆ µ if and only if f (x)a ⊆ ν; (3) ∀ a such that 0 < a ≤ µ(0), fa deﬁned by fa (xa + γ) = f (x)a + δ is an isomorphism of (µ/γ)(a) onto (ν/δ)(a) . Proof. That f exists follows from Theorem 6.5.4. (1) ⇔ (2): If (1) holds, then (2) holds immediately. Suppose that (2) holds. Since xµ(x) ⊆ µ, f (x)µ(x) ⊆ ν by hypothesis and since f (x)ν(f (x)) ⊆ ν, xν(f (x)) ⊆ µ by hypothesis. Thus µ(x) = a implies ν(f (x)) ≥ a and ν(f (x)) = b implies µ(x) ≥ b. Hence µ(x) = ν(f (x)). Thus ν = f (µ) since f is one-to-one. (2) ⇔ (3): Suppose that (2) holds. Now xa + γ = ya + γ ⇔ (−y + x)a ⊆ γ ⇔ f (−y + x)a ⊆ f (γ) ⇔ (−f (y) + f (x))a ⊆ f (γ) ⇔ f (x)a + f (γ) = f (y)a + f (γ) ⇔ f (x)a + δ = f (y)a + δ ⇔ fa (xa + γ) = fa (ya + γ). Thus fa is single-valued and one-to-one. Now xa + γ ∈ (µ/γ)(a) ⇔ xa ⊆ µ ⇔ f (x)a ⊆ ν ⇔ f (x)a + δ ∈ (ν/δ)(a) ⇔ fa (xa + γ) ∈ (ν/δ)(a) . Thus fa maps (µ/γ)(a) onto (ν/δ)(a) since f maps µ∗ onto ν ∗ and so f (x)a + δ is arbitrary in (ν/δ)(a) . Now fa ((xa + γ) +(ya + γ)) = fa ((x + y)a + γ)) = f (x + y)a + δ = (f (x) + f (y))a + δ = (f (x)a + δ) + (f (y)a + δ)) = fa (xa + γ) + fa (ya + γ). Hence fa is an isomorphism. Conversely, suppose (3) holds. Then fa has domain (µ/γ)(a) and image (ν/δ)(a) . Hence (2) holds since xa ⊆ µ ⇔ xa + γ ∈ (µ/γ)(a) ⇔ f (x)a + δ ∈ (ν/δ)(a) ⇔ f (x)a ⊆ ν. We now illustrate Theorem 6.5.9 and other results. Example 6.5.10. Let G = x ⊕ y , where ◦(x) = p and ◦(y) = p3 , p a prime. Let H = x + py . Deﬁne the fuzzy subset µ of G by µ(z) = 1 if z ∈ H and µ(z) = 12 otherwise. Then µ is a fuzzy subgroup of G. The fuzzy subgroups 1 x x µx and µy are such that µx (0) 2 = 1, yµ (z) =1 2 if z ∈ x \ 20 , µ (z) y= 0 y otherwise, µ (z) = 1 if z ∈ p y , µ (z) = 2 if z ∈ y \ p y , and µ (z) = 0 otherwise. Now µx + µy = µx ⊕ µy by Theorem 1.5.9. Let ν = γ = δ = µx ⊕ µy . Then µ∗ = ν ∗ = γ ∗ = δ ∗ = G. Clearly, Iγ = Iδ and so ∃ an

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isomorphism f of γ ∗ onto δ ∗ such that f (γ) = δ by Theorem 6.5.9. However, (x + py)1 ⊆ µ, but f (x + py)1 ⊆ ν else f (H) ⊆ p2 y and so f (H) has order ≤ p. Hence by Theorem 6.5.9, f (µ) = ν. In fact, any isomorphism f of µ∗ onto ν ∗ such that f (µ) = ν for the same reason. We note that Iµ = Iν . However, by Example 6.1.17, µ is not a direct sum of fuzzy subgroups of G whose supports are cyclic. Thus Theorem 6.5.4 is not contradicted.

6.6 Basic and p-Basic Fuzzy Subgroups The representation of G as a direct sum of cyclic groups yields structure results for G. However, not every group G is a direct sum of cyclic groups. But there exist largest subgroups of G that are direct sums of cyclic groups of order inﬁnity and powers of primes. If these largest subgroups are pure, they are of importance in determining properties of G, [4]. These largest pure subgroups, called basic subgroups, are invariants of p-groups [[4], Theorem 35.2, p. 148]. This yields a system of cardinal numbers as a system of invariants of G, [[4], p. 148]. The theory of p-groups is based to a large extent on basic subgroups. Also the concept of basic subgroups as well as other concepts from Abelian group theory are of importance when carried over to the study of inseparable ﬁeld extensions. Let x ∈ µ∗ . Let µx be the fuzzy subset of G deﬁned by µx (y) = µ(y) for all y ∈ x and µx (y) = 0 for all y ∈ G\ x . Then µx is a fuzzy subgroup of G. Let Ix = {nu |u ∈ Im(µx )\{0}, nu the smallest positive integer such that (µx )u = nu x}. Suppose that µ∗ = ⊕j∈A xj . Let Iµ denote the ordered pair, (∪j∈A {Ixj , o(xj )}, Im(µ)\{0}). In the remainder of this section, G denotes an Abelian group and ν a fuzzy subgroup of G . If f is an isomorphism of G onto G , then f (µ) is the fuzzy subset of G deﬁned by ∀ y ∈ G , f (µ)(y) = µ(x), where y = f (x). This deﬁnition coincides with the usual deﬁnition of a homomorphism applied to µ since f is an isomorphism here. By Theorem 1.2.11, f (µ) is a fuzzy subgroup of G . Suppose that µ∗ = ⊕j∈A xj and ν ∗ = ⊕j∈A yj . By Theorem 6.1.12, µxj = {(nuj xj )uj |uj ∈ µxj (G)\{0}) and

ν yj = {(mvj yj )vj |vj ∈ ν yj (G )\{0})

∀ j ∈ A. We call an fuzzy subgroup γ of G cyclic if and only if ∃ x ∈ G such that γ = xa , where a = γ(x). Deﬁnition 6.6.1. Let µ ∈ F(G) and ν ∈ F(µ). Then ν is said to be basic in µ if (1) ν is a direct sum of cyclic fuzzy subgroups of G whose supports are cyclic of prime power orders or of inﬁnite order;

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159

(2) ν is pure in µ, (3) ∀ a such that 0 ≤ a ≤ µ(0), ∀ xa + ν ∈ µ/ν, ∀ n ∈ N, ∃ yb + ν ∈ µ/ν such that n(yb + ν) = xa + ν. Proposition 6.6.2. Let µ ∈ F(G) and ν ∈ F(µ). Then (3) of Deﬁnition 6.6.1 implies µ∗ /ν ∗ is divisible. Proof. Let x + ν ∗ ∈ µ∗ /ν ∗ and let n ∈ N. Then xa + ν ∈ µ/ν for all a such that 0 ≤ a ≤ µ(0). Thus ∃ yb + ν ∈ µ/ν such that xa + ν = nyb + ν. Hence a = b ≤ ν(−x + ny). Thus ν(−x + ny) > 0 and so −x + ny ∈ ν ∗ . Hence x + ν ∗ = ny + ν ∗ and so µ∗ /ν ∗ is divisible. It follows that (3) of Deﬁnition 6.6.1 is equivalent to the exact same statement with b replaced by a. Proposition 6.6.3. Let µ ∈ F(G) and ν ∈ F(µ). Then (3) of Deﬁnition 6.6.1 holds if and only if ∀ a such that 0 ≤ a ≤ ν(0), µa /νa is divisible. Proof. Suppose that (3) holds. Let a be such that 0 ≤ a ≤ ν(0). Let x + νa ∈ µa /νa and n ∈ N. Then µ(x) ≥ a and so xa ⊆ µ. Thus ∃ ya + ν such that n(ya + ν) = xa + ν and so (ny)a + ν = xa + ν. Hence (−x+ ny)a ⊆ ν. Thus ν(−x + ny) ≥ a. Hence −x + ny ∈ νa . Thus ny + νa = x + νa . Hence µa /νa is divisible. Conversely, suppose that µa /νa is divisible for all a such that 0 ≤ a ≤ ν(0). Let a be such that 0 ≤ a ≤ ν(0). Let xa + ν ∈ µ/ν and n ∈ N. Then xa ⊆ µ and so µ(x) ≥ a. Thus x ∈ µa . Hence x + νa ∈ µa /νa . Thus ∃ y + νa ∈ µa /νa such that n(y + νa ) = x + νa . Hence ny + νa = x + νa . Thus −x + ny ∈ νa . Hence ν(−x + ny) ≥ a. Thus (−x + ny)a ⊆ ν. Hence nya + ν = xa + ν. Thus (3) holds. Theorem 6.6.4. Let µ ∈ F(G) and ν ∈ F(µ). If ν is basic in µ, then ∀ a such that 0 ≤ a ≤ ν(0), νa is basic in µa . Proof. The desired result holds from the Propositions 6.4.13, 6.6.3 and Theorem 1.5.10. Theorem 6.6.5. Let µ ∈ F(G). Suppose that µ has the sup property. Let ν ∈ F(µ). If ν is basic in µ, then ν ∗ is basic in µ∗ . Proof. By Proposition 6.6.2, µ∗ /ν ∗ is divisible. Now ν = ⊕j∈A γj , where γj ∈ F(µ) and γj is cyclic. Thus ν ∗ = ⊕j∈A γj ∗ by Theorem 1.5.9 and γj ∗ is cyclic ∀ j ∈ A. Since ν is pure in µ, we have by Proposition 6.4.15 and Proposition 6.4.16 that ∀ n ∈ N, nν = ν ∩ nµ implies nν ∗ = (nν)∗ = (ν ∩ nµ)∗ = ν ∗ ∩ nµ∗ , i.e., ν ∗ is pure in µ∗ . Let p denote a ﬁxed prime. Deﬁnition 6.6.6. Let µ ∈ F(G) and ν ∈ F(µ). Then ν is said to be p-pure in µ if ∀ L-singletons xa ⊆ ν with a > 0, ∀ k ∈ N, ∀ ya ⊆ µ, pk (ya ) = xa implies that ∃ za ⊆ ν such that pk (za ) = xa .

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Deﬁnition 6.6.7. Let µ ∈ F(G) and ν ∈ F(µ). Then ν is said to be p-basic in µ if (1) ν is a direct sum of cyclic fuzzy subgroups whose supports are cyclic of order a power of p or of inﬁnite order; (2) ν is p-pure in µ; (3) ∀ a such that 0 < a ≤ µ(0), ∀ xa + ν, ∀ k ∈ N, ∃ yb + ν such that pk (xa + ν) = yb + ν. Theorem 6.6.8. Let µ ∈ F(G) and ν ∈ F(µ). (1) Then (3) of Deﬁnition 6.6.7 implies µ∗ /ν ∗ is p-divisible; (2) ν is p-pure in µ if and only if νa is p-pure in µa ∀ a such that 0 < a ≤ µ(0); (3) Part (3) of Deﬁnition 6.6.7 holds if and only if µa /νa is p-divisible ∀ a such that 0 < a ≤ µ(0); (4) If ν is p-basic in µ, then νa is p-basic in µa ∀a such that 0 < a ≤ µ(0). (5) Suppose that µ has the sup property. If ν is p-basic in µ, then ν ∗ is p-basic in µ∗ . Proof. The proofs are similar to those of basic fuzzy subgroups.

We let Z denote the additive group of integers and Z(pn ) a cyclic group of order pn , n ∈ N. We recall that for H a p-basic subgroup of G, H = n ⊕∞ n=0 Kn , where K0 = ⊕m0 Z and Kn = ⊕mn Z(p ), n = 1, 2, . . . , and {m0 , m1 , . . . , mn , . . .} is a system of invariants for G, [[4], p. 148]. Theorem 6.6.9. Let µ ∈ F(G) and ν ∈ F(µ) be such that ν is p-basic in µ. (1) ∪a∈A {m0a , m1a , . . . , mna , . . .} is a system of invariants for µ, where {m0a , m1a , . . . , mna , . . .} is the system of invariants determined by νa in µa and A = {a | 0 ≤ a ≤ ν(0)}. (2) Suppose that µ has the sup property. Then {m0 , m1 , . . . , mn , . . .} is a system of invariants for µ, where {m0 , m1 , . . . , mn , . . .} is the system determined by ν ∗ in µ∗ . Proof. The proof is immediate from Theorem 6.6.8 (4) and (5) and the preceding comments. Deﬁnition 6.6.10. Let µ ∈ F(G) and χ ∈ FS(µ). Then χ is said to be pindependent in µ if ∀ x1 , . . . , xk ∈ χ∗ and ∀ ai ∈ {a|0 ≤ a ≤ χ(xi )} for i = 1, . . . , k and ∀ r ∈ N, n1 (x1 )a1 + . . . + nk (xk )ak ⊆ pr µ (ni xi = 0, i = 1, . . . , k) implies pr | ni for i = 1, . . . , k. χ is called a p-basis of µ if χ is p-independent in µ and µ = pµ + χ . Proposition 6.6.11. Let µ ∈ F(G) and χ ∈ FS(µ). Then χ is p-independent in µ if and only if χ∗ is p-independent in µ∗ . Proof. Suppose that χ is p-independent in µ. Let x1 , . . . , xk ∈ χ∗ . Suppose that n1 x1 + . . . + nk xk ∈ pr µ∗ (ni xi = 0, i = 1, . . . , k). Then n1 x1 + . . . +

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nk xk = pr x for some x ∈ µ∗ . Let a = a1 ∧ . . . ∧ ak ∧ µ(x), where ai = χ(xi ) for i = 1, . . . , k. Then n1 (x1 )a + . . . + nk (xk )a = pr xa ⊆ pr µ. Thus pr | ni for i = 1, . . . , k. Conversely, suppose that χ∗ is p-independent in µ∗ . Let x1 , . . . , xk ∈ χ∗ . Suppose that n1 (x1 )a1 + . . . + nk (xk )ak ⊆ pr µ (ni xi = 0, i = 1, . . . , k). Then (n1 x1 + . . . + nk xk )a ⊆ pr µ, where a = a1 ∧. . .∧ak > 0. Thus (pr µ)(n1 x1 + . . . + nk xk ) ≥ a > 0. Hence ∃ x ∈ G such that n1 x1 + . . . + nk xk = pr x. Thus (pr µ)(pr x) > 0 and so µ(x) > 0. Hence x ∈ µ∗ and so pr x ∈ pr µ∗ . Thus pr | ni for i = 1, . . . , k. Deﬁnition 6.6.12. Let µ ∈ F(G) and χ ∈ FS(µ). Then χ is said to be maximally p-independent in µ if χ is p-independent in µ and η ∈ FS(µ) such that η is p-independent in µ and χ ⊂ η. Proposition 6.6.13. Let µ ∈ F(G) and χ ∈ FS(µ). Then χ is maximally p-independent in µ if and only if χ∗ is a p-basis for µ∗ and ∀ x ∈ χ∗ , χ(x) = µ(x). Proof. Suppose that χ is maximally p-independent in µ. Then χ∗ is pindependent in µ∗ by Proposition 6.6.11. Suppose that χ∗ is not a p-basis of µ∗ . Then ∃ y ∈ µ∗ \χ∗ such that χ∗ ∪{y} is p-independent in µ∗ . Thus by Proposition 6.6.11, η ∈ FS(µ), where η = µ on χ∗ ∪{y} and η(z) = 0∀ z ∈ G\(χ∗ ∪ {y}) is p-independent in µ. However, this contradicts the maximality of χ. Conversely, suppose that χ∗ is a p-basis of µ∗ and ∀ x ∈ χ∗ , χ(x) = µ(x). Suppose that χ is not maximal. Then ∃ η ∈ FS(µ) such that χ ⊂ η and η is p-independent in µ. Since χ = µ on χ∗ , χ∗ ⊂ η ∗ . Since η ∗ is necessarily p-independent in µ, this contradicts the maximality of χ∗ . Thus χ is maximal. It follows easily that if χ is a p-basis of µ, then χ is maximally pindependent in µ. However, the following example shows that the converse is not true. Example 6.6.14. Let G = x , where x has order p2 , p a prime. Deﬁne the fuzzy subset µ of G by µ(z) = 1 if z ∈ px and µ(z) = 12 if z ∈ G\ px . Then µ is a fuzzy subgroup of G and µ∗ = G. Now (pµ)(px) = µ(x) = 12 . Thus (pµ)(0) = 1, (pµ)(z) = 12 if z ∈ px \ 0 , and (pµ)(z) = 0 if z ∈ G\ px . If χ is a p-basis of µ, then χ∗ = {z} for some z ∈ G\ px . Now µ(z) = 12 . Thus µ ⊃ pµ + χ since µ(px) = 1, but (pµ + χ)(px) = ∨{(pµ)(u) ∧ χ (v) | px = u + v} = ∨{(pµ)(pix) ∧ χ (p(1 − i)x) | 0 ≤ i < p} ≤ 12 . Thus µ does not have a p-basis even though p2 µ∗ = 0 . Now χ is maximal p-independent in µ since χ∗ is a p-basis for µ∗ . Lemma 6.6.15. Let µ ∈ F(G). ∀ a ∈ (0, µ(0)], pr (µa ) ⊆ (pr µ)a . If either µ has the sup property or µ∗ is torsion-free, then pr (µa ) = (pr µ)a . Proposition 6.6.16. Let µ ∈ F(G) and χ ∈ FS(µ). (1) If χ is p-independent in µ, then χa is p-independent in µa ∀ a such that 0 ≤ a ≤ µ(0).

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(2) Suppose that µ has the sup property or µ∗ is torsion-free. If χa is pindependent in µa ∀ a such that 0 < a ≤ µ(0), then χ is p-independent in µ. Proof. (1) Let x1 , . . . , xk ∈ χa . Suppose that n1 x1 + . . . + nk xk ∈ pr (µa ) (ni xi = 0, i = 1, . . . , k). Then x1 , . . . , xk ∈ χ∗ and n1 x1 + . . . + nk xk ∈ pr µ∗ . By Proposition 6.6.11, pr | ni for i = 1, . . . , k. (2) Let x1 , . . . , xk ∈ χ∗ . Suppose that n1 (x1 )a1 + . . . + nk (xk )ak ⊆ r p µ (ni xi = 0, i = 1, . . . , k). Then (n1 x1 + . . . + nk xk )a ⊆ pr µ, where a = a1 ∧ . . . ∧ ak . By Lemma 6.6.15, (pr µ)a = pr (µa ). Thus n1 x1 + . . . + nk xk ∈ pr (µa ). Since also x1 , . . . , xk ∈ χa , pr |ni for i = 1, . . . , k. Proposition 6.6.17. Let µ ∈ F(G). Suppose that µ has the sup property or µ∗ is torsion-free. Let χ ∈ FS(µ) be such that χ(x) = µ(x) ∀ x ∈ χ∗ . If χa is a p-basis of µa ∀ a such that 0 < a ≤ µ(0), then χ is maximally p-independent in µ. Proof. By Proposition 6.6.16(2), χ is p-independent in µ. Suppose that χ is not maximally p-independent in µ. Then there exists η ∈ FS(µ) such that χ ⊂ η and η is p-independent in µ. By Proposition 6.6.16(1), ηa is p-independent in µa for all a such that 0 < a ≤ µ(0). Since χ∗ ⊂ η ∗ , there exists a such that χa ⊂ ηa . However, this contradicts the maximality of χa . The following example shows that the converse of Proposition 6.6.17 does not hold. Example 6.6.18. Let G = Z(p∞ ) and let x ∈ G have order p. Deﬁne the fuzzy subset µ of G by µ(z) = 1 if z ∈ x and µ(z) = 12 otherwise. Then µ is a fuzzy subgroup of G and µ∗ = G. Now 0G is maximally p-independent in µ since ∅ is a p-basis of µ∗ . However, µ1 = x and (0G )1 = ∅ is not a p-basis of x . Note also that 0G is not a p-basis of µ. Let µ ∈ L(G). If χ is maximally p-independent in µ, we determine in the following to what extent χ is a p-basic L-subgroup of µ. Proposition 6.6.19. Let µ ∈ F(G) and χ ∈ FS(µ). If χ is p-independent in µ, then χ is independent in µ. Proof. χ is p-independent in µ ⇔ χ∗ is p-independent in µ∗ by Proposition 6.6.11. If χ∗ is p-independent in µ∗ , then χ∗ is independent in µ∗ . Now χ∗ is independent in µ∗ ⇔ χ is independent in µ. Theorem 6.6.20. Let µ ∈ F(G) and χ ∈ FS(µ). Then χ is independent in µ if and only if χ = ⊕xa ∈Sχ xa , where Sχ = {xa | x ∈ χ∗ , χ(x) = a}. Proof. See Theorem 6.2.5.

Theorem 6.6.21. Let µ ∈ F(G) and χ ∈ FS(µ). If χ is p-independent in µ, then χ = ⊕xa ∈Sχ xa , where Sχ = {xa | x ∈ χ∗ , χ(x) = a}.

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Proof. The desired result follows by Proposition 6.6.19 and Theorem 6.6.20. Lemma 6.6.22. Let µ ∈ F(G) and χ ∈ FS(µ). If χ is independent in µ, then χa = χa for all a such that 0 ≤ a ≤ µ(0). Proof. If a > ∨{χ(x) | x ∈ χ∗ }, then the result follows easily. Since χa ⊆ χa , χa ⊆ χa . Let x ∈ χa . Then χ (x) ≥ a. Now χ = ⊕xS ∈Sχ xS by Theorem 6.6.20. Thus x has an unique representation, x = n1 x1 + . . . + nk xk , where xi ∈ χ∗ . Thus a ≤ a1 ∧ . . . ∧ ak , where χ(xi ) = ai for i = 1, . . . , k. Hence x ∈ χa . Thus χa ⊆ χa . Theorem 6.6.23. Let µ ∈ F(G) and χ ∈ FS(µ). If χ is p-independent in µ, then χ is p-pure in µ. Proof. Since χ is p-independent in µ, χa is p-independent in µa ∀ a ∈ {a|0 ≤ a ≤ µ(0)} by Proposition 6.6.16. Thus χa = χa is p-pure in µa by Lemma 6.6.22 and [4]. Hence χ is p-pure in µ. Let µ ∈ F(G). We also note that if χ ∈ FS(µ) is maximal p-independent, ∗ then χ∗ = χ is a p-basic subgroup of µ∗ since χ∗ is a p-basis of µ∗ by Proposition 6.6.13. Theorem 6.6.24. Let µ ∈ F(G) and χ ∈ FS(µ). (1) Suppose that χ is p-independent in µ. If χ is a p-basic in µ, then χ is a p-basis of µ. (2) Suppose that µ is ﬁnite-valued. If χ is a p-basis of µ, then χ is p-basic in µ. Proof. (1) Let xa ⊆ µ. Then by (3) of Deﬁnition 6.6.1, ∃ ya ⊆ µ such that xa + χ = pya + χ . Thus µ = pµ + χ . (2) By Theorem 6.6.21 and Lemma 6.6.22, it suﬃces to show that µa / χa is divisible ∀ a ∈ {a|0 ≤ a ≤ µ(0)}. Since µ is ﬁnite-valued, p(µa ) = (pµ)a by Lemma 6.6.15 and (pµ + χ)a = (pµ)a + χa since pµ and χ are necessarily ﬁnite-valued. Thus from µ = pµ + χ , it follows that µa = p(µa ) + χa by Lemma 6.6.22. Hence by Proposition 6.6.16, χa is a p-basis of µa . Thus µa / χa is divisible ∀ a such that 0 ≤ a ≤ µ(0). Theorem 6.6.25. Let µ ∈ F(G) and ν ∈ F(µ). If ν is p-basic in µ, then χ is a p-basis of µ, where ν = ⊕j∈A (xj )aj and χ is the fuzzy subset of G deﬁned by χ(z) = aj if z = xj and χ(z) = 0 otherwise. Proof. Now ν ∗ = ⊕j∈A xj . Thus χ∗ is a p-basis of ν ∗ . Since ν ∗ is p-pure in µ∗ , χ∗ is p-independent in µ∗ . Since µ∗ /ν ∗ is divisible by Proposition 6.6.2, χ∗ is a p-basis of µ∗ . Thus χ is maximally p-independent in µ by Proposition 6.6.13. Now ν = χ . Hence χ is a p-basis of µ by Theorem 6.6.24(1). The proof of Theorem 6.6.25 also shows that if ν is p-basic in µ, then µ = ν on ν ∗ .

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Theorem 6.6.26. Let µ ∈ F(G). If µ∗ has a unique p-basic subgroup, then µ has a unique p-basic subgroup. Proof. Let ν and γ be p-basic subgroups of µ. Then ν ∗ and γ ∗ are p-basic subgroups of µ∗ and so ν ∗ = γ ∗ . Now ν = χ and γ = η for p-bases χ and η of µ by Theorem 6.6.25. By Proposition 6.6.13, χ = µ on χ∗ and η = µ on η ∗ . Thus ν = µ on ν ∗ and γ = µ on γ ∗ . But ν ∗ = γ ∗ and so ν = γ. Let µ ∈ F(G). Criteria for µ∗ to contain exactly one p-basic subgroup can be found in [[4], Theorem 35.3, p. 149]. Theorem 6.6.27. Let µ ∈ F(G). Then the following assertions hold. (1) Suppose that µ∗ has a unique p-basic subgroup and that µ is ﬁnitevalued. If χ and η are p-bases of µ, then Iχ = Iη . (2) Suppose that pe µ = θ for some positive integer e. If χ and η are p-bases of µ, then Iχ = Iη . Proof. For assertion (1), χ and η are p-basic subgroups of µ by Theorem 6.6.24(2). Thus χ = η . For assertion (2), since pe µ = 0G , we have that µ ∗ = χ = η . In either case, let f be the identity map on χ . Then f (χ) ∗ ∗ ∗ ∗ = η . Now χ = χ = ⊕ xj and so η = η = ⊕ f (xj ) and Iχ = Iη by Theorem 6.5.5. Suppose that G is torsion. Suppose that G has a ﬁnite p-basis. Let G and K be subgroups of G, K a subgroup of K ∩ G , and H a subgroup of G . Suppose that K ⊂ K and G ⊂ G. We note that it is impossible for K and H to be p-basic subgroups of G and for K and H to be p-basic subgroups of G : Suppose otherwise. Then we that have G = pG + K , G = pG + H, G = pG + K ⊃ pG + K (since G has a ﬁnite p-basis), and G = pG + H. Thus G = pG + G = pG + pG + K = pG + K , a contradiction. Theorem 6.6.28. Suppose that G is torsion. Let µ ∈ F(G). Suppose that µ∗ has a ﬁnite p-basis and that µ is ﬁnite-valued. Let ν, γ ∈ F(µ). If ν and γ are p-basic in µ, then ν(G)\{0} = γ(G)\{0}. Proof. Let B = ν(G)\{0} and C = γ(G)\{0}. Since µ = ν on ν ∗ and µ = γ on γ ∗ , B ⊆ µ(G) and C ⊆ µ(G). Since µ is ﬁnite-valued, ν and γ are ﬁnite-valued. Suppose that B = C. Then either ∃ a ∈ B such that a ∈ / C or ∃ b ∈ C such that b ∈ / B, say a exists. Then a < ν(0). Since B and C are ﬁnite sets, either (1) ∃ largest b ∈ C such that b < a or (2) a < ∧{b | b ∈ C}. Suppose that (1) holds. Then ∃ smallest v ∈ B, v ∈ / C, such that b < v ≤ a. Now ∃ smallest u ∈ C such that b < v < u since γ(0) > v. Thus γv = γu and νv ⊃ νu . By Theorem 6.6.8(4), γv and νv are p-basic in µv and γu and νu are p-basic in µu . However, as noted above, this is impossible since µv ⊃ µu , where the latter inclusion holds since v, u ∈ µ(G). Suppose that (2) holds. Let u = ∧{b | b ∈ C}. Then γa = γu and νa ⊃ νu and we have a contradiction as just argued.

References

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Theorem 6.6.29. Suppose that G is torsion. Let µ ∈ F(G). Suppose that µ∗ has a ﬁnite p-basis and that µ is ﬁnite-valued. Let ν, γ ∈ F(µ). If ν and γ are p-basic in µ, then ∃ isomorphism f of ν ∗ onto γ ∗ such that f (ν) = γ and Iν = Iγ . Proof. By Theorem 6.6.25, ν and γ are ﬁnitely generated. Now ν = ⊕j∈A (xj )aj , where ν ∗ = ⊕j∈A xj and γ = ⊕m∈B (ym )bm , where γ ∗ = ⊕m∈B ym by Theorem 6.6.25. By Theorem 6.6.25, |A| = |B| < ∞. By Theorem 6.6.28, ν(G)\{0} = γ(G)\{0} and these sets are ﬁnite. Thus ν(G)\{0} = {a1 , . .. , an } = γ(G)\{0}, where a1 < . . . < an . Since ν is ﬁnite-valued, νa = ⊕j∈A (xj )aj a ∀ a ∈ ν(G). Now A = A1 ∪ . . . ∪An , where j ∈ Ai if and only if aj = ai . Thus νai = ⊕j∈Ai xj ⊕ ⊕j∈Ai+1 ∪...∪An xj and ⊕j∈Ai xj νai /νai+1 and ⊕j∈Ai+1 ∪...∪An xj νai+1 for i = 1, . . . , n − 1. Thus νai (νai /νai+1 )⊕νai+1 and similarly γai (γai /γai+1 ) ⊕γai+1 . Now there exists an isomorphism gi of νai onto γai for i = 1, . . . , n by Theorem 6.6.4. Since νai+1 is a direct summand of νai and γai+1 is a direct summand of γai and these groups are ﬁnitely generated, there exists an isomorphism fi of νai /νai+1 onto γai /γai+1 such that gi = fi ⊕ gi+1 , i = 1, . . . , n − 1. Let fn = gn . Then f = f1 ⊕. . .⊕fn is an isomorphism of ν ∗ = νa1 = (νa1 /νa2 )⊕ . . . ⊕(νan−1 /νan )⊕νan onto γ ∗ = γa1 = (γa1 /γa2 ) ⊕ . . . ⊕ (γan−1 /γan ) ⊕ γan such that f (ν) = f (γ). Hence Iν = Iγ by Theorem 6.5.5.

References 1. S. C. Cheng and Z. Wang, Divisible T L-subgroups and pure T L-subgroups, Fuzzy Sets and Systems 78 (1996) 387-393. 139 2. P. S. Das, Fuzzy subgroups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264-269. 3. J. K. Deveney and J. N. Mordeson, Inseparable extensions and primary abelian groups, Arch. Math. 33 (1979) 538-545. 139 4. L. Fuchs, Inﬁnite Abelian Groups, Vol. 36 Academic Press, New York 1970. 152, 154, 158, 160, 164 5. W. Gu and T. Lu, The properties of fuzzy divisible groups, Fuzzy Sets and Systems 56 (1993) 195-198. 139 6. L. Y. Kulikov, On the theory of abelian groups of arbitrary cardinality, Mat. Sbornik 16 (1945) 129-162 (in Russian). 7. T. Lu and W. Gu, Abelian fuzzy group and its properties, Fuzzy Sets and Systems 64 (1994) 415-420. 8. D. S. Malik and J. N. Mordeson, Fuzzy subgroups of abelian groups, Chinese J. Math. (1991) 129-145. 139 9. D. S. Malik, J. N. Mordeson, and P. S. Nair, Fuzzy generators and fuzzy direct sums of abelian groups, Fuzzy Sets and Systems 50 (1992) 193-199. 139 10. M. A. A. Mishref, Primary fuzzy subgroups, Fuzzy Sets and Systems 112 (2000) 313-318. 11. J. N. Mordeson, Modular extensions and abelian groups, Arch. Math. 36 (1981) 13-20. 139 12. J. N. Mordeson, Generating properties of fuzzy algebraic structures, Fuzzy Sets and Systems 55 (1993) 107-120.

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13. J. N. Mordeson, Invaraiants of fuzzy subgroups, Fuzzy Sets and Systems 63 (1994) 81-85. 139 14. J. N. Mordeson, Fuzzy group subalgebras, J. Fuzzy Math. 3 (1995) 69-81. 139 15. J. N. Mordeson, Fuzzy group subalgebras II, J. Fuzzy Math. 3 (1995) 885-897. 139 16. J. N. Mordeson and D, S, Malik, Fuzzy Commutative Algebra, World Scientiﬁc, Inc., 1998. 139 17. J. N. Mordeson and M. K. Sen, Basic fuzzy subgroups, Inform. Sci. 82 (1995) 167-179. 139 18. M. Petrich, Introduction to Semigroups, Charles E. Merrill Publishing Co., Columbus Ohio, 1973. 19. S. Sebastian and S. B. Sandar, Commutative L-fuzzy subgroups, Fuzzy Sets and Systems 68 (1994) 115-121. 20. F. I. Sidky and M. A. Mishref, Divisible and pure fuzzy subgroups, Fuzzy Sets and Systems 34 (1990) 377-382. 139, 151 21. W. Waterhouse, The structure of insepsarable ﬁeld extensions, Trans. Amer. Math. Soc. 211 (1975) 39-56. 139

7 Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups

In Chapter 6, a necessary and suﬃcient condition for a fuzzy subgroup to be a weak direct sum of fuzzy subgroups was obtained by employing known structure theorems for Abelian groups. In [31] and [23], the problem of expressing a normal fuzzy subgroup of a direct product of groups as a fuzzy direct product of certain subgroups of a group G was examined. This is essential in order to obtain structure theorems involving fuzzy subgroups of a group G and fuzzy subgroups of it subgroups. In this chapter, we ﬁrst we introduce the notion of the fuzzy direct product of fuzzy subgroups of subgroups of a group and employ these ideas to the problem in group theory of obtaining conditions under which a group G can be expressed as the direct product of its normal subgroups. We extend the deﬁnition of weak direct sum introduced in Chapter 6 for Abelian groups to the general case and establish a one-to-one correspondence between these two fuzzy direct products. Thus the relevant results of fuzzy direct products can be applied to fuzzy weak direct products. ∈ I} be a We let G denote a group throughout this chapter. Let {µi | i collection of fuzzy subsets of G. Then the fuzzy subsets ∩i∈I µi and i∈I µi of µ )(x) = ∧{µ (x) | i ∈ I} and ( µ G are deﬁned by ∀x ∈ G, (∩i∈I i i∈I i )(x) = i I}, where all but a ﬁnite ∨{∧{µi (xi ) | i ∈ I} | x = i∈I xi , xi ∈ G, i ∈ number of the xi are the identity in the product i∈I xi . Let µ be a fuzzy subgroup of G. The chain of subgroups {µt | t ∈ Im(µ)} is called the chain of level subgroups of µ. Recall that µ∗ = {x ∈ G | µ(x) = µ(e)} is a subgroup of G. Two fuzzy subgroups of G are said to be equivalent if they have the same chain of level subgroups. It is easy to check that two fuzzy subgroups of G are equal if and only if they are equivalent and they have the same images. Recall that fuzzy subgroup µ of G is normal if and only if its chain of level subgroups forms a normal chain of G. A fuzzy subgroup of G is said to be subnormal if its chain of level subgroups forms a subnormal chain of G. It is clear that every normal fuzzy subgroup is a subnormal fuzzy subgroup. However the converse is not true, [[28], No. 355, p. 142]. John Mordeson, Kiran R. Bhutani, Azriel Rosenfeld: Fuzzy Group Theory, StudFuzz 182, 167– 200 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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7.1 Fuzzy Direct Products We now consider fuzzy direct products. This work is mainly by Alkhamees [5]. The following deﬁnition is a slight modiﬁcation of the notion of direct product of fuzzy subgroups given in Chapter 1. Deﬁnition 7.1.1. Let {Gi | i ∈ I} be a collection of normal subgroups of G and µi a fuzzy subgroup of Gi ∀i ∈ I. Let µ be a fuzzy subgroupof G. Then µ ⊗ is called the fuzzy direct product of the µi , denoted by µ = i∈I µi , if the following conditions hold: ⊗ (1) G = i∈I Gi ; (2) µ(e) = µi (e) ∀i ∈ I; (3) if x ∈ G, x = e, then µ(x) = ∧{µi (xi ) | xi = e, i ∈ I}, where x = i∈I xi is the unique representation of x as a product of elements of G. Example 7.1.2. Let G = {e, a, b, c} be the Klein 4-group, where e is the identity of G. Let H = a and K = b. Let µ, ν, ρ be the fuzzy subgroups of G, H, K, respectively, deﬁned as follows: µ(b) = µ(c) = s0 , µ(a) = s1 , µ(e) = s2, ν(a) = s1 , ν(e) = s2 , ρ(b) = s0 , ρ(e) = s2 , where 0 ≤ s0 < s1 < s2 ≤ 1. Clearly, G = H ⊗ K. Now µ(a) = ν(a), µ(b) = ρ(b), and µ(c) = ν(a) ∧ ρ(b). Thus µ = ν ⊗ ρ. Theorem 7.1.3. Let {Gi | i ∈ I} be a collection of subgroups of G and let µi be a fuzzy subgroup of Gi ∀i ∈ I. A fuzzy subgroup µ of G is the fuzzy direct product of the µi , i ∈ I, if and only if the following conditions hold: the restriction of µ to Gi ∀i ∈ I; (1) µi is ⊗ (2) µs = i∈I (µi )s ∀s ∈ Im(µ). ∗ Proof. Assume conditions (1) and (2) hold. ∧Im(µ) and si = ⊗Let s = ⊗ ∧Im(µi ) ∀i ∈ I. Then clearly G = µs∗ = i∈I (µi )si = i∈I Gi , where the ∗ second equality follows from the fact that si ≥ s and so (µi )s∗ = (µi )si ∀i ∈ I. Let s ∈ Im(µ) and z ∈ G be such that µ(z) = s. Let z = zk1 zk2 ...zkn be the unique representation of z as a product of of Gi , where zki = e for elements ⊗ all i = 1, ..., n. Since z ∈ µs and µs = i∈I (µi )s , zki ∈ (µki )s , i = 1, ..., n. Thus µki (zki ) ≥ s for i = 1, ..., n. Let t = ∧{µki (zki ) | i = 1, ..., n} and J = {i . | µki (zki ) = t, i = 1, ..., n}. If J = {1, ..., n}, let k = zki and h = zzk−1 i Then µ(k) ≤ µ(z) ≤ µ(zh−1 ) = µ(k). Thus s = µ(z) = µ(k) = t. If J = {1, ..., n}, then clearly z can be expressed as z = hk, where h = i∈I\J zki and k = j∈J zkj . We next show that µ(z) = µ(k). If µ(hk) ≥ µ (h) , then µ(k) = µ((h−1 h)k) ≥ µ(h) ∧ µ(hk) ≥ µ(h).

7.1 Fuzzy Direct Products

169

µ(k) = µ((h−1 h)k) ≥ µ(h) ∧ µ(hk) ≥ µ(h). Thus µ(k) = µ(h), a contradiction. Hence µ(hk) < µ(h). Now µ(hk) ≥ µ(h) ∧ µ(k) and so µ(hk) ≥ µ(k). Also µ(k) = µ((h−1 h)k) ≥ µ(h) ∧ µ(hk) = µ(hk). Hence we have that µ(z) = µ(hk) = µ(k) = t. Therefore, if z ∈ G, z = e, then µ(z) = ∧{µi (zi ) | zi = e, i ∈ I},

µ(z) = ∧{µi (zi ) | zi = e, i ∈ I},

where z = i∈I zi is the unique representation of z as a product of elements of Gi . Thus µ is a fuzzy direct product of the µi . Conversely, suppose µ is a fuzzy direct product of the µi . Then clearly µi is the restriction of µ to Gi ∀i ∈ I. Also, by the deﬁnition of fuzzy direct ⊗ product, it follows easily that µs = i∈I (µi )s ∀s ∈ Im(µ). Corollary 7.1.4. Let µ be a fuzzy direct product of µi , i ∈ I. Then Im(µ) = ∪i∈I Im(µi ). Deﬁnition 7.1.5. Let µ be a fuzzy subgroup of G and let {µi | i ∈ I} be a collection of fuzzy subgroups of G. If µ is a fuzzy direct product of the µi , we use the notation PIµ = {µi | i ∈ I}. Let ν be a fuzzy subgroup of G and let {νi | i ∈ I} be a collection of fuzzy subgroups of G such that ν is a fuzzy direct product of the νi . Then we say that PIµ is equivalent to PIν if the following conditions are satisﬁed: (1) µi is equivalent to νi ∀i ∈ I. (2) ∀s ∈ Im(µ), (µi )s = (νi )t ∀i ∈ I, where t = ∧{fi (s) | s ∈ Im(µi ), i ∈ I} and fi is a bijection of Im(µi ) onto Im(νi ) ∀i ∈ I. Example 7.1.6. Let G, H, K and µ, ν, ρ be deﬁned as in Example 7.1.2. Let µ , ν , ρ be the fuzzy subgroups of G, H, K, respectively, deﬁned as follows: µ (a) = µ (ab) = s0 , µ (b) = s1 , µ (e) = s2 ν (a) = s0 , ν (e) = s2 ν (a) = s0 , ν (e) = s2 ρ (b) = s1 , ρ (e) = s2 , where 0 ≤ s0 < s1 < s2 ≤ 1. Then clearly µ = ν ⊗ ρ, µ = ν ⊗ ρ , ν is equivalent to ν and ρ is equivalent to ρ . However, µ is not equivalent to µ since their level sets diﬀer. Theorem 7.1.7. Suppose that µ and ν are fuzzy subgroups of G such that µ and ν are fuzzy direct products of the collections of fuzzy subgroups {µi | i ∈ I} and {νi | i ∈ I}, repectively. Then µ is equal (equivalent) to ν if and only if PIµ is equal (equivalent) to PIν .

170

7 Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups

Proof. Suppose µ is equivalent to ν. Let f be the bijection of Im(µ) onto Im(ν) such that µs = νf (s) for all s ∈ Im(µ). Let fi denote the restriction of f to Im(µi ). Suppose s ∈ Im(µj ), where j ∈ I. Then s ∈ Im(µ). By Theorem 7.1.3, µs =

⊗

(µi )s , νf (s) =

i∈I

Since µs = νf (s) ,

⊗

⊗ (νi )f (s). i∈I

(µi )s =

i∈I

⊗

(νi )f (s).

i∈I

Since µs ∩Gj = (µj )s and µs = νf (s) and so µs ∩Gj = νf (s) ∩Gj = (νj )f (s) , we have (µj )s = (νj )f (s) ∀s ∈ Im(µj ). Hence µj is equivalent to νj ∀j ∈ I. The other can easily be arrived at from the fact that ⊗ condition of Deﬁnition7.1.5 ⊗ (µ ) = µ = ν = (ν )f (s). Therefore, PIµ is equivalent to PIν . i s s i f (s) i∈I i∈I µ Conversely, suppose PI is equivalent to PIν . Let fi be the bijection of Im(µi ) onto Im(νi ) such that (µi )s = (νi )fi (s) for all i ∈ I. Let f be the function from Im(µ) into Im(ν) deﬁned as follows: ∀s ∈ Im(µ), f (s) = ∧{fi (s)|s ∈ Im(µi ), i ∈ I}. Then ∀s ∈ Im(µ), (µi )s = (µi )si for some si ∈ Im(µi ) and hence (µi )s = (νi )fi (si ) since µi is equivalent to νi . By condition (2) of Deﬁnition 7.1.5, we have (µi )s = (νi )f (s) . Therefore, (νi )f (s) = (νi )fi (si ) ∀i ∈ I. Also, ∀s ∈ Im(µ), µs =

⊗ ⊗ (µi )s = i ∈ I(µi )si i∈I

⊗ ⊗ i ∈ I(νi )f (s) = (νi )fi (si ) = i∈I

= νf (s) . Hence if f (s) = f (t), then νf (s) = νf (t) . Since µs = νf (s) and µt = νf (t) , µs = µt . Thus s = t. Therefore, f is one-one. Let t ∈ Im(ν). Then t ∈ Im(νi ) for some i ∈ I. Suppose s = ∨{si ∈ Im(µi ) | fi (si ) = t, i ∈ I}. Then clearly f (s) = t and so f is bijective. Since µs = νf (s) ∀s ∈ Im(µ), µ is equivalent to ν. Lemma 7.1.8. Let µ be a fuzzy subgroup of G. Suppose that µ is subnormal (normal) and that µ is a fuzzy direct product of fuzzy subgroups µi , i ∈ I. Then µi is subnormal (normal) ∀i ∈ I. Proof. Suppose µ is subnormal (normal). Let s, t ∈ Im(µj ) be such that s < t and such that there is no k ∈ Im(µj ) such that s < k < t. Then s, t ∈ Im(µ). Suppose s ∈ Im(µ) is such that s < s and such that there is no k ∈ Im(µ) such that s < k < s . This implies s ≤ t. Further,

7.1 Fuzzy Direct Products

µs =

171

⊗ (µi )s i∈I

µs =

⊗ (µi )s i∈I

=

⊗

(µi )s ⊗ (µj )t .

i∈I,i =j

Since µ is subnormal (normal), we have µs µs (µs G). Thus (µj )t (µj )s ((µj )t G). Therefore, µj is subnormal (normal). We next consider direct products of fuzzy subgroups of composition (principal) length. Deﬁnition 7.1.9. Let µ be a fuzzy subgroup of G. If Im(µ) is ﬁnite, let l(µ) = |Im(µ)| − 1. Then l(µ) is called the length of µ. Deﬁnition 7.1.10. A normal fuzzy subgroup µ of G is called a fuzzy subgroup of principal length if the chain of level subgroups of µ forms a principal (chief ) series. Deﬁnition 7.1.11. A subnormal fuzzy subgroup µ of G is called a fuzzy subgroup of composition length if the chain of level subgroups of µ forms a composition series of G. It is known that the composition length and the principal length are unique, [[28], pp. 121-126 and p.142]. Furthermore, if the composition length exists, it is maximal of all lengths of subnormal fuzzy subgroups of G and if the principal length exists, it is maximal of all lengths of normal fuzzy subgroups of G. It is known that a group has a composition (principal) series if and only if it satisﬁes the subnormal (normal) chain conditions. By this result and by the fact that all subnormal (normal) chains of such groups have equivalent composition (principal) series as reﬁnements [[28], Theorem 7.9 p. 125 and p. 142], the following theorem holds. Theorem 7.1.12. A group G has a composition (principal) series if and only if every subnormal (normal) fuzzy subgroup of G is of ﬁnite length. We need the following result of group theory [[28], Theorem 7.36, p. 142]. Recall that a principal factor of a group G is a factor group H/K where H and K are consecutive terms of a principal series of G. Theorem 7.1.13. Suppose that G has principal series. Let H, K be normal subgroups of G with K ⊂ H. Then H/K is a principal factor of G if and only if H/K is a minimal normal subgroup of G/K.

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7 Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups

Proposition 7.1.14. Let H, K be normal subgroups of G such that H ∩ K = {e}. Let N be a normal subgroup of H. Then N ⊗ K H ⊗ K. Proof. Let hk ∈ H ⊗ K and h k ∈ N ⊗ K. Since K G, hk(h k )(hk)−1 = hkh k k −1 h−1 = hh k1 k k −1 h−1 = hh h−1 k2 ∈ N ⊗ K, where k1 , k2 ∈ K.

Theorem 7.1.15. Suppose that G has a composition series and µ is a subnormal fuzzy subgroup of G such that µ is a fuzzy direct product of fuzzy subgroups µi of subgroups Gi of G, i = 1, ..., n. Then µ is of composition length if and only if µi is of composition length and Im(µi ) ∩ Im(µj ) = {µ(e)} ∀i = j, i, j = 1, ..., n. Proof. We prove the theorem by induction on n. It suﬃces ⊗ to prove it for n = 2 and then repeat the same argument by taking H = i∈I Gi (I = {1, 2, ..., n − 1}) and K = Gn . Thus suppose that H and K are subgroups of G and that ν and ρ are fuzzy subgroups of H and K, respectively, such that µ = ν ⊗ ρ. Assume ν and ρ are of composition length and Im(ν) ∩ Im(ρ) = {µ(e)}. Suppose µ is not of composition length. Then the chain of level subgroups of µ can be reﬁned to obtain another subnormal chain. Hence there exists a a subgroup N of G such that µt N µs , where s, t ∈ Im(µ) with s < t and there is no r in Im(µ) with s < r < t. Since Im(ν) ∩ Im(ρ) = {µ(e)}, ν(e) = µ(e) = ρ(e). Since H ∩ K = {e}, it is clear that the only common element in the domains of ν and ρ is e. Let x ∈ N \µt . Then s ≤ µ(x) < t and so µ(x) = s. Suppose neither ν nor ρ are deﬁned on x. Then x = hk, where h ∈ νs ⊆ H and k ∈ ρs ⊆ K. Now s = µ(x) = ν(h) ∧ ρ(k) and ν(h) = ρ(k). Say s = µ(x) = ν(h) < ρ(k). Since there is no r in Im(µ) such that s < r < t, there is no such r in Im(ρ) by Corollary 7.1.4. Thus k ∈ ρt ⊆ µt ⊆ N and so k ∈ K ∩ N. Since x ∈ N, h ∈ N. Thus h ∈ H ∩ N. Suppose ν is deﬁned on x. Then ρ is not deﬁned on x. Hence µ(x) = s = ν(x). Thus x ∈ νs ∩ N ⊆ H ∩ N. Suppose x ∈ µt . Then x = hk, where h ∈ νt ⊆ H and k ∈ ρt ⊆ K. Since νt , ρt ⊆ µt ⊆ N, h ∈ H ∩ N and k ∈ K ∩ H. Hence N = (H ∩ N ) ⊗ (H ∩ N ). Let N = H ∩ N and N = K ∩ N. Also, µs = νs ⊗ ρs , µt = νt ⊗ ρt and Im(ν) ∩ Im(ρ) = {µ(e)} imply that either νt = N or ρt = N . Suppose νt = N . Deﬁne ν of H as follows: ∀x ∈ H, the fuzzy subset r x ∈ N \νt ν (x) = ν(x) otherwise, where r ∈ [0, 1] with t < r < s and s , t ∈ Im(ν) such that νs = νs and νt = νt . Then it follows that ν is subnormal fuzzy subgroup of H. However, this contradicts the assumption that ν is of composition length since l(ν ) > l(ν). Therefore, µ is of composition length. Conversely, assume that µ is of composition length, s, t ∈ Im(ν) with s < t and there is no r in Im(ν) with s < r < t. Also, let s ∈ Im(µ) with s < s

7.1 Fuzzy Direct Products

173

and suppose there are no r in Im(µ) such that s < r < s . Then µs = νs ⊗ ρs and µs = νs ⊗ ρs . If ρs = ρs , then using the fact that ρ is a subnormal fuzzy subgroup of K (Lemma 7.1.8) and Proposition 7.1.14, it follows easily that νs ⊗ ρs is a proper normal subgroup of µs . Since µs is a proper normal subgroup of νs ⊗ρs , (νs ⊗ρs )/µs is a proper normal subgroup of µs /µs . This contradicts the fact µs /µs ∼ = νs /νt is simple since µ is of composition length. Therefore, ρs = ρs and hence µs /µs ∼ = νs /νt . Thus νs /νt is a simple group and hence ν is of composition length. Similarly, it can be shown that ρ is of composition length. Finally, let s ∈ Im(ν) ∩ Im(ρ), s = µ(e), and s ∈ Im(µ) with s < s . Suppose there is no r in Im(µ) such that s < r < s . Then (νs ⊗ ρs )/µs is a nontrivial normal subgroup of µs /µs . This contradicts the fact that µs /µs is a simple group. Therefore, Im(ν) ∩ Im(ρ) = {µ(e)}. Theorem 7.1.16. Suppose that G has a principal series and µ is a normal fuzzy subgroup of G such that µ is a fuzzy direct product of fuzzy subgroups µi of subgroups Gi of G, i = 1, · · · , n. Then µ is of principal length if and only if µi is of principal length and Im(µi ) ∩ Im(µj ) = {µ(e)} ∀i = j, i, j = 1, . . . , n. Proof. As in the previous theorem, it suﬃces to prove the theorem for n = 2. Suppose H, K are subgroups of G and that ν and ρ are fuzzy subgroups of H and K, respectively. If ν and ρ are of principal length, then by a similar argument as used in the proof of the ﬁrst part of the previous theorem, it can be shown that µ is of principal length. Conversely, assume that µ is of principal length. Let s, t ∈ Im(ν) with s < t and such that there is no r ∈ Im(ν) such that s < r < t. Let s ∈ Im(µ). Then µs = νs ⊗ ρs and µs = νs ⊗ ρs = νt ⊗ ρs . Suppose ρs = ρs . Since νs = H ∩ µs and µs , H G, νs G and since ρ is a normal fuzzy subgroup of K by Lemma 7.1.8, ρs K. Thus by using a similar method to that used in Proposition 7.1.14, it follows that νs ⊗ ρs G. Hence (νs ⊗ ρs )/µs is a normal subgroup of G/µs . However, since (νs ⊗ ρs )/µs is a proper subgroup of µs /µs , we have a contradiction of the fact that µs /µs is a minimal normal subgroup of G/µs by Theorem 7.1.13 since µ is of principal length. Therefore, ρs = ρs and so µs /µs ∼ = νs /νt . Since (H ⊗ ρs )/µs is a subgroup of G/µs and µs /µs is a subgroup of G/µs , µs /µs is a minimal normal subgroup of (H ⊗ ρs )/µs ∼ = H/νt and µs /µs ∼ = νs /νt . Thus it follows that νs /νt is a minimal normal subgroup of H/νt . By Theorem 7.1.13, it follows that ν is fuzzy subgroup of H of principal length. Similarly, it follows that ρ is fuzzy subgroup of K of principal length. Finally, let s ∈ Im(ν) ∩ Im(ρ), s = µ(e), and s ∈ Im(µ) with s < s . Suppose there is no r in Im(µ) such that s < r < s . Then it follows as above that (νs ⊗ νs )/µs is a non-trivial normal subgroup of µs /µs . This contradicts the fact that µs /µs is a minimal normal subgroup of G/µs . Therefore, Im(ν) ∩ Im(ρ) = {µ(e)}. We next obtain necessary and suﬃcient conditions for a group G to be a direct product of its normal subgroups.

174

7 Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups

Theorem 7.1.17. Suppose that G has a principal series. Then G is a direct product of its normal subgroups G1 , . . . , Gn if and only ifthere exists n i = 1l(µi ) a fuzzy subgroup µ of G of principal length such that l(µ) = and Im(µi ) ∩ Im(µj ) = {µ(e)}, i = j, where µi is the restriction of µ to Gi , i, j = 1, ..., n. Proof. We prove the theorem by induction on n. Suppose H and K are normal subgroups of G and µ is a fuzzy subgroup of G of principal length such that l(µ) = l(ν) + l(ρ) and Im(ν) ∩ Im(ρ) = {µ(e)}, where ν, ρ are the restrictions of µ to H, K, respectively. Let x ∈ H ∩ K. Then µ(x) = ν(x) = ρ(x). Hence µ(x) = µ(e) and so x ∈ µ∗ . However, µ∗ = {e} since µ is of principal length. Thus H ∩K = {e} and so H ⊗K is a normal subgroup of G. Since l(µ) = l(ν)+ l(ρ), |(Im(ν)\{µ(e)}) ∪ (Im(ρ)\{µ(e)})| = |Im(µ)\{µ(e)}|. Suppose Im(µ) = {s0 , s1 , . . . , sn }, µsi = µi , νsi = νi and ρsi = ρi , where 0 ≤ s0 < s1 < . . . < sn ≤ 1. Deﬁne the fuzzy subset δ of H ⊗ K as follows: ∀x ∈ H ⊗ K, δ(x) = si if x ∈ µi \µi+1 , i = 0, 1, ..., n. Then since µi G ∀i = 1, 2, . . . , n, it follows that δ is a normal fuzzy subgroup of H ⊗ C with δsi = νi ⊗ ρi ∀i = 1, 2, ..., n. However, clearly the length of δ is equal to the length of µ. Therefore, G = H ⊗ K and δ = µ, otherwise G has a normal fuzzy subgroup of length greater than the length of µ which is impossible by the assumption that µ is of principal length. Conversely, suppose that G = H ⊗ K, and H = H0 H1 . . . Hh = {e} K = K0 K1 . . . Kk = {e} are principal series of H and K, respectively. By a similar argument as used in the proof of Proposition 7.1.14, it follows that H ⊗ Ki G ∀i = 1, . . . , k. Clearly, Hj G ∀j = 1, . . . , h. Thus G = (H ⊗ K0 ) (H ⊗ K1 ) . . . (H ⊗ Kk−1 ) H0 H1 . . . Hh = {e} is a normal series of G. By Theorem 7.1.13, we have that Ki /Ki+1 is a minimal normal subgroup of K/Ki+1 . However, since (H ⊗ Ki )/(H ⊗ Ki+1 ) is isomorphic to Ki /Ki+1 and (H ⊗ K)/(H ⊗ Ki+1 ) is isomorphic to K/Ki+1 , we have that (H⊗Ki )/(H⊗Ki+1 ) is a minimal normal subgroup of (H⊗K)/(H⊗Ki+1 ) and by Theorem 7.1.13 again, we have that it is a principal factor. Hence the H ⊗ Ki are terms of the principal series of G, i = 0, 1, . . . , k − 1. Let µ be a fuzzy subgroup of G which has the above series as its chain of level subgroups. Suppose Im(µ) = {s0 , s1 , . . . , sk , sk+1 , . . . , sk+h }, where 0 ≤ s0 < s1 < . . . < sk+h ≤ 1. Set ν = µ|H and ρ = µ|K . Then clearly µ is of principal length, l(µ) = l(ν) + l(ρ) and Im(ν) ∩ Im(ρ) = {µ(e)}.

7.1 Fuzzy Direct Products

175

The following theorem can be proved in a similar manner as the previous theorem. Theorem 7.1.18. Suppose that G has a composition series. Then G is a direct product of its normal subgroups G1 , ..., Gn if and only ifthere exists a n fuzzy subgroup µ of G of composition length such that l(µ) = i=1 l(µi ) and Im(µi ) ∩ Im(µj ) = {µ(e)} ∀i = j, where µi are the restrictions of µ to Gi , i = 1, ..., n . Theorem 7.1.19. Suppose G has a composition (principal) series and µ is a fuzzy subgroup of G of composition (principal) length. Then µ is fuzzy direct product of its restrictions n µ1 , ..., µn to the normal subgroups G1 , ..., Gn of G if and only if l(µ) = i=1 l(µi ) and Im(µi ) ∩ Im(µj ) = {µ(e)} ∀i = j, i, j = 1, . . . , n. Proof. It is suﬃcient to prove the theorem for n = 2. Let H and K be subgroups of G and let ν and ρ be fuzzy subgroups of H and K, respectively. Assume that µ = ν ⊗ ρ, s ∈ Im(ν) ∩ Im(ρ), s = µ(e), and s , t, r are elements of Im(µ), Im(ν), Im(ρ), respectively, with s , t, r < s. Assume also that there are no k1 , k2 , k3 in Im(µ), Im(ν), Im(ρ), respectively, such that s < k1 < s , s < k2 < t and s < k3 < r. Then as in the proof of Theorem 7.1.15, we obtain that if µ is of composition length or as in the proof of Theorem 7.1.16 that if µ is of principal length, νs = νt or ρs = ρr . However, this contradicts the above assumption. Therefore, Im(ν) ∩ Im(ρ) = {µ(e)}. Since µ = ν ⊗ ρ, Im(µ) = Im(ν) ∪ Im(ρ). However, Im(ν) ∩ Im(ρ) = {µ(e)}. Therefore, l(µ) = l(ν) + l(ρ). The converse follows as in the proof of Theorem 7.1.17 in the case that µ is of principal length. A similar technique for the case that µ is of composition length can be used. We now consider the correspondence between fuzzy direct products and fuzzy weak direct products. Deﬁnition 7.1.20. Let {νi | i ∈ I} be a collection of subnormal fuzzy subgroups of G. Then a subnormal fuzzy subgroup ν of G is said to be a fuzzy weak direct product of the νi if the following conditions hold: (1) ν = i∈I νi , (2) (νj ∩ ( i∈I,i =j νi ))(x) = 0 ∀x ∈ G, x = e. Deﬁnition 7.1.20 is a generalization of [[25], Deﬁnition 4.1, p. 140] to the noncommutative case. Using a similar technique as in the proof of [[25], Theorem 4.6, p. 142] and [[25], Corollary 4.7, p. 142], this corollary can be generalized to obtain the following result. Theorem 7.1.21. Let ν be a subnormal fuzzy subgroup of G and {νi | i ∈ I} a collection of subnormal fuzzy subgroups of G such that νi ⊆ ν ∀i ∈ I. Then ν is a fuzzy weak direct product of νi if and only if νi (e) = ν(e) ∀i ∈ I and

176

7 Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups

νs =

⊗

(νi )s ∀s ∈ (0, ν(e)].

i∈I

Supposethat {Gi | i ∈ I} is a collection of normal subgroups of G and ⊗ that G = i∈I Gi . Suppose µ and ν are subnormal fuzzy subgroups of G. µ Let PI be deﬁned as previously. Let WIν denote the set {νi | i ∈ I}, where the νi are subnormal fuzzy subgroups of G such that ν is a fuzzy weak direct product of the νi , (νi )∗ is a subgroup of Gi and νi ⊆ ν ∀i ∈ I. Now ∀i ∈ I, deﬁne the fuzzy subset µi of G by ∀x ∈ G, µi (x) = µi (x) if x ∈ Gi and µi (x) = 0 otherwise. Let (PIµ ) denote the set of all the µi and let (WIν ) denote the set of all the νi such that νi is the restriction of νi to Gi , i ∈ I. Let A = {PIµ | µ ∈ F(G) and µ is subnormal}, B = {WIν | ν ∈ F(G) and ν is subnormal}, A = {(PIµ ) | µ ∈ F(G) and µ is subnormal}, and B = {(WIν ) | ν ∈ F(G) and ν is subnormal}. The following results show how a fuzzy weak direct product can be produced from a fuzzy direct product and vice versa and how through this process a one-to-one correspondence between A and B can be established. Hence the relevant results of this section can be applied to WIν . Proposition 7.1.22. Let PIµ ∈ A and WIν ∈ B. Then (PIµ ) ∈ B, WIν ∈ A, ((PIµ ) ) = PIµ and ((WIν ) ) = WIν . Proof. By Lemma 7.1.8, µi is a subnormal fuzzy subgroup of Gi ∀i ∈ I. Since (µi )s = (µi )s ∀s ∈ (0, µ(e)], it follows easily that µi is a subnormal fuzzy subgroup of G ∀i ∈ I. By Theorem 7.1.3, the µi are the restrictions of µ to Gi . Also, µi (x) = µ (x) ∀x ∈ Gi and so µi ⊆ µ ∀i ∈ I. By Theorem 7.1.3, µs =

⊗ i∈I

(µi )s =

⊗

(µi )s ∀s ∈ (0, µ(e)].

i∈I

Thus by Theorem 7.1.21, we have that (PIµ ) ∈ B. Clearly, µi is a subnormal fuzzy subgroup of Gi and (µi ) = µi ∀i ∈ I. Therefore, ((PIµ ) ) = PIµ . Since the νi are the restrictions of νi to Gi and (νi )s = (νi )s ∀s ∈ (0, ν(e)], clearly the νi are fuzzy subnormal subgroups of Gi . Also, by Theorem 7.1.21, ∈ I. Hence νi (e) = ν(e) ∀i ∈ I. Let x ∈ Gi , x = e. By Theorem νi (e) = ν(e) ∀i ⊗ ⊗ 7.1.21, νs = i∈I (νi )s ∀ s ∈ (0, ν(e)] and so it follows that ν ∗ = i∈I (νi )∗ and ifνi (x) = 0, then νi (x) = νi (x) = ν(x) ∀i ∈ I. If νi (x) > 0, then since ⊗ νs = i∈I (νi )s ∀ s ∈ (0, ν(e)], it follows that νi (x) = νi (x) = ν(x) ∀i ∈ I. Therefore, νi (x)= ν(x) ∀x ∈ Gi . Hence the νi are the restrictions of ν to ⊗ ⊗ ∗ ∗ i ∈ I(νi )s ∀s ∈ (0, ν(e)], we have that Gi . Since ν = i∈I (νi ) and νs = ⊗ νs = i∈I (νi )s ∀s ∈ Im(ν). Therefore, WIν ∈ A by Theorem 7.1.3. Finally, since the (νi ) are subnormal fuzzy subgroups of G and (νi ) = νi , it follows that ((WIµ ) ) = WIµ . We now have the following theorem.

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177

Theorem 7.1.23. Deﬁne φ from A into B and ψ from B into A as follows: ∀PIµ ∈ A, φ(PIµ ) = (PIµ ) and ∀WIν ∈ B, ψ(WIµ ) = (WIµ ) . ∀PIµ ∈ A, φ(PIµ ) = (PIµ ) and ∀ WIν ∈ B, ψ(WIµ ) = (WIµ ) .Then φ and ψ are one-one correspondences such that φ ◦ ψ is the identity of B and φ ◦ ψ is the identity of A.

7.2 Fuzzy p-groups In this section, we are interested in the work in [17]. We introduce the notion of a fuzzy p-subgroup and prove that a fuzzy subgroup can be written as the intersection of its minimal fuzzy p-subgroups. Recall that we denote the identity of a group G by e, the order of G by o(G), the order of x in G by o(x), and the greatest common divisor of integers n1 , ..., nt by (n1 , ..., nt ). Deﬁnition 7.2.1. Let µ be a fuzzy subgroup of a group G. If p is prime, then µ is called a fuzzy p-subgroup of G if F Oµ (x) is a power of p for all x ∈ G. It follows that the order of a fuzzy p-subgroup is a power of p if the order is ﬁnite. In Chapter 6 , we introduced the notion of a p-primary fuzzy subgroup of an Abelian group by using fuzzy singletons. We showed that there is a unique maximal p-primary fuzzy subgroup contained in a fuzzy subgroup for each prime p. However, the notion of a fuzzy p-subgroup as deﬁned in Chapter 6 is diﬀerent from the one used here. Proposition 7.2.2. Let µ be a fuzzy subgroup of a group G such that µ∗ is a normal subgroup of G. Then µ is a fuzzy p-subgroup if and only if G/µ∗ is a p-group. Proof. Suppose µ is a fuzzy p-subgroup. Let x ∈ G. Then F Oµ (x) = pt t for some nonnegative integer t. Hence xp ∈ µ∗ . Thus G/µ∗ is a p-group. t Conversely, suppose G/µ∗ is a p-group. Then for x ∈ G, xp ∈ µ∗ for some t nonnegative integer t and so µ(xp ) = µ(e). Hence µ is a fuzzy p-subgroup of G by Proposition 1.6.5. Now we consider the order of the intersection of fuzzy subgroups. Proposition 7.2.3. Let µ and ν be fuzzy subgroups of a group G such that µ(e) = ν(e). Let O(µ) = mpr11 ...prt t and O(ν) = nps11 ...pst t for ri , si ∈ N, i = 1, ..., t. Suppose that (m, pi ) = (n, pi ) = 1, where (m, n) = 1 and the pi are distinct primes for i = 1, ..., t. Then O(µ∩ν) = mnpu1 1 ...put t , where ui = ri ∨si for all i = 1, ..., t.

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Proof. Let k = mnpu1 1 ...put t . Then (µ ∩ ν)(xk ) = µ(xk ) ∧ ν(xk ) = µ(e) ∧ ν(e) = (µ ∩ ν)(e) for all x ∈ G since O(µ) and O(ν) both divide k. Thus O(µ∩ν)|k. By Theorem 2.3.22, O(µ) and O(ν) both divide O(µ ∩ ν). However, k is the least common multiple of O(µ) and O(ν). Hence we have the desired conclusion. Corollary 7.2.4. The intersection of a ﬁnite number of fuzzy p-subgroups of a group G such that their images of e are equal is again a fuzzy p-subgroup of G. Corollary 7.2.5. Let µ be a fuzzy subgroup of a group G. If there exists a minimal fuzzy p-subgroup ν of G containing µ such that ν(e) = µ(e), then ν is unique. We denote the unique minimal fuzzy p-subgroup in Corollary 7.2.5 by µ(p) if it exists. However, µ(p) does not exist in general as can be seen by the following example. Example 7.2.6. Let p be prime. Deﬁne the fuzzy subgroup µ of Z by ∀x ∈ Z, t0 if x = 0, µ(x) = t1 otherwise, where t0 > t1 . For all n ∈ N, deﬁne a fuzzy subgroup νn of Z by ∀x ∈ Z, t0 if x ∈ pn Z, νn (x) = t1 otherwise. Then ∀n ∈ N, F Oνn (x) = pj if x ∈ pn−j Z\pn−j+1 Z for j = 1, ..., n and F Oνn (x) = p0 if x ∈ pn Z. Hence ν1 ⊃ ν2 ⊃ ... is an inﬁnite descending chain of fuzzy p-subgroups of Z containing µ and such that ∩∞ i=1 νn = µ. Thus it follows that µ(p) does not exist since µ is not a fuzzy p-subgroup. In [21], a fuzzy subgroup µ of G is deﬁned to be a fuzzy Sylow p-subgroup of G if |Im(µ)| ≤ 2 and µ∗ is a Sylow p-subgroup of G. Let µ be a fuzzy Sylow p-subgroup of a ﬁnite nilpotent group G and ν a fuzzy subgroup of G with ν∗ = {e} . Then ν∗ is the Sylow p-subgroup Gp of G. However, (ν(p) )∗ is the pcomplement of G, where ν(p) is the minimal fuzzy p-subgroup of G containing ν (cf., Proposition 7.2.9 and Lemma 7.2.8) and G/(ν(p) )∗ is isomorphic to Gp . Thus the notion of a fuzzy Sylow p-subgroup of [21] is somewhat diﬀerent from the notion of a minimal fuzzy p-subgroup presented here. However, these two notions are related as follows: (i) ∩{ν(q) | q is a prime and q = p} is a fuzzy Sylow p-subgroup of G if the number of the images of ν is less than or

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179

equal to 2, (ii) the minimal fuzzy p-subgroup µ(p) of G containing µ is such that (µ(p) )∗ = G. In the remainder of the section, we are concerned with the problem of whether or not a fuzzy subgroup µ can be written as the intersection of its minimal p-subgroups. Lemma 7.2.7. Let µ be a fuzzy subgroup of a group G. Let x, y ∈ G. If (F Oµ (x), F Oµ (y)) = 1 and xy = yx, then µ(xy) = µ(x) ∧ µ(y). Proof. Let F Oµ (y) = m. Then µ(xy) ≤ µ((xy)m ) = µ(xm y m ) = µ(xm ) by Lemma 2.1.2. However, µ(xm ) = µ(x) by Proposition 1.6.11. Hence µ(xy) ≤ µ(x). Also, µ(xy) ≤ µ(y) holds similarly. Thus µ(xy) = µ(x) ∧ µ(y). Lemma 7.2.8. Let µ be a fuzzy subgroup of a group G satisfying the conditions (1), (2) and (3) for some prime p. (1) F Oµ (x) is ﬁnite for all x ∈ G. (2) Hp = {x ∈ G | (F Oµ (x), p) = 1} and Kp = {x ∈ G | F Oµ (x) is a power of p} are subgroups of G. (3) For all x, y ∈ G, there exist expressions x = x1 x2 = x2 x1 and y = y1 y2 = y2 y1 of x and y respectively, as in Theorem 1.6.16, such that x1 , y1 ∈ Hp , x2 , y2 ∈ Kp , x1 y2 = y2 x1 and x2 y1 = y1 x2 . Deﬁne a fuzzy subset νp of G by νp (x) = µ(x2 ), where x2 is the element in Theorem 1.6.16, i.e., where x = x1 x2 = x2 x1 , F Oµ (x1 ) = m and F Oµ (x2 ) = pt with F Oµ (x) = mpt and (m, p) = 1. Then νp is a fuzzy p-subgroup of G containing µ. Proof. By Theorem 1.6.16, νp is well-deﬁned. For x, y ∈ G, let x = x1 x2 = x2 x1 and y = y1 y2 = y2 y1 be expressions of x and y, respectively, as in condition (3). Let z = xy. By conditions (2) and (3), z = z1 z2 = z2 z1 is an expression of z = xy, where z1 = x1 y1 and z2 = x2 y2 . Thus νp (xy) = µ(z2 ) = µ(x2 y2 ) ≥ µ(x2 ) ∧ µ(y2 ) = νp (x) ∧ νp (y). Clearly, νp (x−1 ) ≥ νp (x) for all x ∈ G. Therefore, νp is a fuzzy subgroup of G. Clearly, νp is a fuzzy p-subgroup of G containing µ. Proposition 7.2.9. Let νp be the fuzzy subgroup deﬁned in Lemma 7.2.8. Then νp = µ(p) . In particular, if F Oµ (x) is a power of p for x ∈ G, then µ(x) = µ(p) (x).

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7 Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups

Proof. Let ξ be a fuzzy p-subgroup of G such that µ ⊆ ξ ⊆ νp . Let x ∈ G. Suppose that F Oµ (x) = mpt and (m, p) = 1. If t = 0 then, µ(xm ) = µ(e). Now µ ⊆ ξ. Thus ξ(xm ) = ξ(e) and so ξ(x) = µ(e) = νp (x) since νp is a fuzzy psubgroup and by Proposition 1.6.5. If m = 1, then µ(x) ≤ ξ(x) ≤ νp (x) = µ(x) and so ξ(x) = νp (x). Finally, let x = x1 x2 = x2 x1 be an expression of x as in Lemma 7.2.8. Then ξ(x1 ) = µ(e) and ξ(x2 ) = νp (x2 ) = µ(x2 ) by the above cases. Hence ξ(x) = ξ(x1 x2 ) = µ(x2 ) = νp (x). Theorem 7.2.10. Let µ be a fuzzy subgroup of a group G such that the conditions of Lemma 7.2.8 hold for all primes p. Then µ = ∩µ(p) . Proof. Let x ∈ G and let F Oµ (x) = pr11 ...prt t , where the pi are distinct primes and the ri are positive integers, i = 1, ..., t. Let qi = F Oµ (x)/(pri i ), i = 1, ..., t. Then (q1 , ..., qt ) = 1. Hence there exist k1 , ..., kt ∈ Z such that q1 k1 +...+qt kt = 1. Now F Oµ (xqi ki ) divides pri i for all i = 1, ..., t and the F Oµ (xqi ki ) are pairwise coprime. Therefore, µ(x) = µ(xq1 k1 ...xqt kt ) = µ(xq1 k1 ) ∧ ... ∧ µ(xqt kt ) by Lemma 7.2.7 = µ(p1 ) (xq1 k1 ) ∧ ... ∧ µ(pt ) (xqt kt ) by Proposition 7.2.9 = µ(p1 ) (x) ∧ ... ∧ µ(pt ) (x) ≥ (∩(p) µ)(x). Thus µ ⊇ ∩µ(p) . Clearly, µ ⊆ µ(p) . Hence we have the desired conclusion.

Lemma 7.2.11. Let µ be a fuzzy subgroup of a group G. If either (1) or (2) below hold, then the conditions of Lemma 7.2.8 hold for all primes p. (1) G is a ﬁnite nilpotent group. (2) G is an Abelian group and F Oµ (x) is ﬁnite for all x ∈ G. Proof. If G is ﬁnite and nilpotent, then G is the direct product of its Sylow subgroups. Thus the remainder of the proof follows easily. Corollary 7.2.12. Let µ be a fuzzy subgroup of a group G. If G is an Abelian group with F Oµ (x) < ∞ for all x ∈ G or if G is a ﬁnite nilpotent group, then µ = ∩µ(p) . That Theorem 7.2.10 does not hold in general can be seen by the following example. Example 7.2.13. Let D3 = a, b|a3 = b2 = e, ba = a2 b be the dihedral group with 6 elements. Deﬁne the fuzzy subgroup µ of D3 by ∀x ∈ D3 , t0 if x = e, µ(x) = t1 otherwise, where t0 > t1 . Then µ(2) is as follows:

7.2 Fuzzy p-groups

µ(2) (x) =

181

t0 if x ∈ a , t1 otherwise,

since µ(2) (b2 ) = µ(2) ((ab)2 ) = µ(2) ((ba)2 ) = µ(2) (e). Let p be an odd prime and ν a fuzzy subgroup of G. Then ν(bp ) = ν(e) ⇒ ν(b) = ν(e). Also, ν(ap ) = ν(e) ⇒ p = 3 ⇒ ap = e. Furthermore, ν((ab)3 ) = ν(ab) and so ν((ab)3 ) = ν(e) ⇒ ν(ab) = ν(e). Similarly, ν((ba)3 ) = ν(e) ⇒ ν(ba) = ν(e). Thus µ(3) (x) = µ(e) ∀x ∈ D3 . In fact, it follows that for all odd primes p, µ(p) is a trivial fuzzy subgroup, i.e., µ(p) (x) = µ(e) for all x ∈ D3 . Since F Oµ (a) is not a power of 2, µ is not a fuzzy 2-subgroup. Thus ∩µ(p) = µ(2) = µ. Theorem 7.2.14. Let µ be a fuzzy subgroup of G. Suppose G is either Abelian with O(µ) < ∞ or G is a ﬁnite nilpotent group. If n is a positive integer that divides O(µ), then there is a fuzzy subgroup ξ of G containing µ such that O(ξ) = n and ξ(e) = µ(e). Proof. Let O(µ) = pr11 ...prt t , where the pi are distinct primes and the ri are positive integers for i = 1, ..., t. Then for all i = 1, ..., t, µ(pi ) exists by Lemma 7.2.11, µ(pi ) equals νpi (deﬁned in Lemma 7.2.8) by Proposition 7.2.9, and O(νpi ) = pri i by Proposition 2.3.19. We may assume that n = pu1 1 ...pus s with s ≤ t and 1 ≤ ui ≤ ri ,i = 1, ..., s. For all i = 1, ..., s, deﬁne a fuzzy subgroup ξi of G by ∀x ∈ G, µ(e) if pui i divides F Oνpi (x2 ), ξi (x) = νpi (x) otherwise, where x = x1 x2 = x2 x1 is an expression of x as in Lemma 7.2.8. Then ξi (e) = µ(e) and O(ξi ) = pui i since pui i |pri i , i = 1, ..., s. Thus ∩ξi is the desired fuzzy subgroup of G. We now apply our results to direct products of fuzzy subgroups. It was shown in [31] that every fuzzy subgroup of the direct product of certain groups can be written as the direct product of its projections. This result was improved in [3] and [23] (see Corollary 7.2.17 and Corollary 7.2.16). In this section, we extend their results by using the notion of orders of fuzzy subgroups. Let µi be a fuzzy subgroup of a group Gi , i = 1, ..., n, where n is a positive integer. The direct product µ1 ⊗ ... ⊗ µn is the fuzzy subgroup of the direct product G1 ⊗ ... ⊗ Gn deﬁned by (µ1 ⊗ ... ⊗ µn )(x1 , ..., xn ) = µ1 (x1 ) ∧ ... ∧ µn (xn ). Let µ be a fuzzy subgroup of the direct product of G1 ⊗ ... ⊗ Gn . For all i = 1, ..., n, the projection µi of µ on Gi is the fuzzy subgroup of Gi deﬁned by µi (x) = µ(e1 , ..., ei−1 , x, ei+1 , ..., en )

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7 Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups

for all x ∈ Gi . The following result extends [[3], Theorem 5.3, p. 99] by replacing the assumption (Gi , Gj ) = 1 by (O(µi ), O(µj )) = 1 for all distinct i, j ∈ {1, ..., n} . Theorem 7.2.15. Let Gi be a group for i = 1, ..., n, where n ∈ N. Let µ be a fuzzy subgroup of the direct product of the Gi , i = 1, ..., n. Suppose O(µ) is ﬁnite. Let µi be the projection of µ on Gi for all i = 1, ..., n. If (O(µi ), O(µj )) = 1 for all distinct i, j ∈ {1, ..., n}, then µ = µ1 ⊗ ... ⊗ µn . Proof. Let (x1 , ..., xn ) ∈ G1 ⊗ ... ⊗ Gn . Let i, j ∈ {1, ..., n} , i < j. Then (F Oµi (xi ), F Oµj (xj )) = 1 since (O(µi ), O(µj )) = 1. Hence (F Oµ (e1 , ..., ei−1 , xi , ei+1 , ..., en ), F Oµ (e1 , ..., ej−1 , xj , ej+1 , ..., en )) = 1. Thus µ(e1 , ...., ei−1 , xi , ei+1 , ..., ej−1 , xj , ej+1 , ..., en ) = µ (e1 , ..., ei−1 , xi , ei+1 , ..., en ) ∧ µ (e1 , ..., ej−1 , xj , ej+1 , ..., en ) = µi (xi ) ∧ µ (j) (xj ) by Lemma 7.2.7. Therefore, µ(x1 , ..., xn ) = µ1 (x1 ) ∧ ... ∧ µn (xn ) by induction on n since the O(µi ) are pairwise co-prime, where i = 1, ..., n.

The following two corollaries follow easily from Theorem 7.2.15. Corollary 7.2.16. [3, 23]. Let µ be a fuzzy subgroup of the direct product of the groups G1 , ..., Gn , where the Gi have pairwise coprime orders for i = 1, ..., n. Then µ can be written as the direct product µ1 ⊗ ... ⊗ µn , where each µi is the projection of µ on Gi . Proof. By Theorem 2.3.17, the order of µi divides the order of Gi for all i ∈ {1, ..., n} . Corollary 7.2.17. [31]. Let µ be a fuzzy subgroup of the direct product of groups G1 , ..., Gn , where the Gi are cyclic groups of distinct prime power orders for i = 1, ..., n. Then µ can be written as the direct product µ1 ⊗ ... ⊗ µn , where every µi is the projection of µ on Gi , i = 1, ..., n. Corollary 7.2.17 can be easily proved by using Theorem 7.2.10. Let pi be the prime factor of o(Gi ) for all i ∈ {1, ..., n} . Then the projection of µ(pi ) can be identiﬁed with µi , i = 1, ..., n, and it follows easily that ∩ni=1 µ(pi ) = µ1 ⊗ ... ⊗ µn .

7.3 Fuzzy Subgroups Having Property ∗

183

7.3 Fuzzy Subgroups Having Property ∗ The notion of a generalized characteristic fuzzy subgroup was introduced in [1], presented in Section 4.2, and a characterization of all ﬁnite cyclic groups in terms of generalized characteristic fuzzy subgroups was given. In this section, the notion of the fuzzy order of an element is used to introduce the notion of what is called property (∗) [16]. This is a generalized and more fuzziﬁed notion of the notion of a generalized characteristic fuzzy subgroup. We characterize all ﬁnite cyclic groups in terms of this generalized notion and give an easy proof for the characterization in [1]. Finally, we suggest an improvement upon a theorem in [8] concerning the normal fuzzy subgroups of a group of square free order. The results of this section are from [16]. Throughout this section, we denote the identity of a group G by e, the order of x in G by o(x), and the greatest common divisor of integers m and n by (m, n). F Oµ (x) is always a divisor of o(x). If µ∗ = {e} , then F Oµ (x) = o(x). However, the two statements o(x) = o(y) and F Oµ (x) = F Oµ (y) are independent of each other, as is illustrated by Example 1.6.4 and the following example. Example 7.3.1. Deﬁne the fuzzy subset µ of Z2 ⊕ Z4 by µ(x) = t0 if x ∈ (0, 2) and µ(x) = t1 otherwise, where t0 > t1 . Then µ is a fuzzy subgroup of Z2 ⊕Z4 and F Oµ ((0, 1)) = F Oµ ((1, 0)) = 2, but o((0, 1)) = 4 and o((1, 0)) = 2. Recall from Section 4.2 that a fuzzy subgroup µ of a ﬁnite group G is called a generalized characteristic fuzzy subgroup (brieﬂy, a GCFS) if o(x) = o(y) implies µ(x) = µ(y) for all x, y in G. We now replace o(x) by F Oµ (x) in this deﬁnition. Deﬁnition 7.3.2. A fuzzy subgroup µ of a group G is said to have property (∗) if F Oµ (x) = F Oµ (y) implies µ(x) = µ(y) for all x, y in G. The fuzzy subgroup µ in Example 1.6.4 has property (∗), but µ is not a GCFS. Proposition 7.3.3. Let µ be a GCFS of a ﬁnite group G and let x, y ∈ G. If o(x) = o(y), then F Oµ (x) = F Oµ (y). Proof. Suppose that o(x) = o(y). Then o(xn ) = o(y n ) and so µ(xn ) = µ(y n ) for all positive integers n. Thus F Oµ (x) = F Oµ (y). Proposition 7.3.4. Let µ be a GCFS of a ﬁnite p-group G and let x, y ∈ G. If F Oµ (x) = F Oµ (y) > 1, then o(x) = o(y).

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7 Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups

Proof. Let F Oµ (x) = F Oµ (y) = pt , where t ∈ N. Then o(x) = pt+m and o(y) = pt+n for some nonnegative integers m and n by Proposition m−n 1.6.5. We may assume that m ≥ n. Since o(xp ) = pt+n = o(y), m−n m−n p t F Oµ (x ) = F Oµ (y) = p by Proposition 7.3.3. However, F Oµ (xp )= F Oµ (x)/(F Oµ (x) , pm−n ) = pt /(pt , pm−n ) by Theorem 1.6.10. Thus m = n since t ≥ 1. The following may be considered a corollary to Theorem 1.6.10. Corollary 7.3.5. If µ is a GCFS of a ﬁnite nilpotent group G, then µ has property (∗). Proof. Let x, y ∈ G. Suppose F Oµ (x) = F Oµ (y). Then by Proposition 1.6.5 and Theorem 1.6.13, F Oµ (xp ) = F Oµ (yp ) for every prime p, where xp and yp denote the p-component of x and y, respectively. Thus µ(xp ) = µ(yp ) by Proposition 7.3.4. Hence µp (xp ) = µp (yp ) for every prime p, where µp denotes the projection of µ on the p-component Gp of G. Therefore, µ(x) = µ(y) by Corollary 7.2.16. We thus see that the property (∗) is a generalized and more fuzziﬁed notion of the notion of GCFS for at least nilpotent ﬁnite groups. This leads to the following question. Does a GCFS of a ﬁnite group always have the property (∗)? We point out that a fuzzy subgroup may neither be a GCFS nor have property (∗). See, for example, Lemma 7.3.11. Lemma 7.3.6. Let µ be a fuzzy subgroup of a group G. Then µ∗ is normal in G if and only if F Oµ (x) = F Oµ (y −1 xy) for all x, y in G. Proof. Suppose µ∗ is normal in G. Let x ∈ G. If F Oµ (x) = ∞, then xn ∈ / µ∗ for all positive integers n. Thus for all y ∈ G, (y −1 xy)n = y −1 xn y ∈ / µ∗ and so F Oµ (y −1 xy) = ∞. If F Oµ (x) = n for some n ∈ N, then xn ∈ µ∗ and so (y −1 xy)n = y −1 xn y ∈ µ∗ . Thus F Oµ (y −1 xy) divides n = F Oµ (x) by Proposition 1.6.5. Hence F Oµ (x) = F Oµ (y −1 xy) for all x, y ∈ G. Conversely, if x ∈ µ∗ , then F Oµ (y −1 xy) = F Oµ (x) = 1. Thus y −1 xy ∈ µ∗ for all y ∈ G. Proposition 7.3.7. Let µ be a fuzzy subgroup of a group G. If µ has property (∗) and µ∗ is normal in G, then µ is normal in G. Proof. The result follows from Lemma 7.3.6.

The assumption in Proposition 7.3.7 that µ∗ is normal in G cannot be removed. This can be seen by Example 7.3.8. The converse of the proposition is not true in general as can be seen by Example 7.3.9. Furthermore, under the assumption in the proposition, if G/µ∗ is ﬁnite, then g(µ) is clearly a

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GCFS, where g is the natural homomorphism of G onto G/µ∗ . However, a level subgroup of µ is not a characteristic subgroup of G in general. For example, µt0 = ab is not a characteristic subgroup of G in Example 1.6.4. Example 7.3.8. Deﬁne the fuzzy subset µ of the dihedral group D3 = a, b|a3 = b2 = e, ba = a2 b by ∀x ∈ G, µ(x) = t0 if x ∈ b and µ(x) = t1 otherwise, where t0 > t1 , Then µ is a fuzzy subgroup of G and µ has property (∗). However, µ∗ is not normal in G and µ is not normal in G. Example7.3.9. Deﬁne the fuzzy subset µ of the Klein 4-group 2 G = a, b|a2 = b2 = (ab) = e by µ(e) = t0 , µ(a) = t1 , and µ(b) = µ(ab) = t2 , where t0 > t1 > t2 . Then µ is a normal fuzzy subgroup of G. Since F Oµ (a) = 2 = F Oµ (b) and µ(a) = µ(b), µ does not have property (∗). Proposition 7.3.10. Let f be a homomorphism of a group G onto a group H and let µ and ν be fuzzy subgroups of G and H, respectively. Then the following properties hold. (1) If µ has property (∗) and µ is f -invariant, then f (µ) has property (∗). (2) If ν has property (∗), then f −1 (ν) has property (∗). Proof. (1) Since µ is f -invariant, (f (µ))(f (x)n ) = µ(xn ) for all integers n. Hence if F Of (µ) (f (x)) = F Of (µ) (f (y)), then F Oµ (x) = F Oµ (y) and so (f (µ))(f (x)) = µ(x) = µ(y) = (f (µ))(f (y)). Thus f (µ) has the property (∗). (2) Since (f −1 (ν))(xn ) = ν(f (xn )) for all integers n, if F Of −1 (ν) (x) = F Of −1 (ν) (y), then F Oν (f (x)) = F Oν (f (y)) and so (f −1 (ν))(x) = ν(f (x)) = ν(f (y)) = (f −1 (ν))(y). Thus f −1 (ν) has the property (∗). Lemma 7.3.11. Let H and K be subgroups of a group G. If x ∈ H\K and y ∈ K\H are such that o(x) = o(y), then there exists a fuzzy subgroup µ of G such that µ is neither a GCFS nor has property (∗). Proof. Deﬁne the fuzzy subset µ of G by ∀z ∈ G, µ(e) = t1 , µ(z) = t2 if z ∈ H\{e}, and µ(z) = t3 otherwise, where t1 > t2 > t3 . Then µ is a desired fuzzy subgroup of G by Deﬁnition 7.3.2 and Proposition 7.3.3 since µ(x) = t2 = t3 = µ(y). Corollary 7.3.12. Let G be a ﬁnite group. If either (1) or (2) holds, then G is cyclic. (1) Every fuzzy subgroup of G is a GCFS. (2) Every fuzzy subgroup µ of G has property (∗) and µ∗ is normal in G. Proof. In the quaternion group Q = a, b|a4 = e, b2 = a2 , ba = a3 b , a ∈ a \ b , b ∈ b \ a , and o(a) = o(b) = 4. Thus if (1) holds, then since every subgroup of G can be realized as a level subgroup of some fuzzy subgroup and since a GCFS of a ﬁnite group must be normal, G is Abelian by Theorem 4.1.10 and Lemma 7.3.11. Also, if (2) holds, then since any subgroup of G can be realized as a level subgroup of some fuzzy subgroup, G is Abelian by Proposition 7.3.7, Theorem 4.1.10, and Lemma 7.3.11. Hence G is cyclic by Lemma 7.3.11.

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Corollary 7.3.12 gives an easy proof for the suﬃciency of the theorem in [1] which states that if G is a ﬁnite group, then G is cyclic if and only if every fuzzy subgroup of G is a GCFS. We now give another characterization of ﬁnite cyclic groups which follows immediately from Corollary 7.3.12 and Theorem 1.7.3. Theorem 7.3.13. Let G be a ﬁnite group. Then G is cyclic if and only if every fuzzy subgroup µ of G has property (∗) and µ∗ is normal in G. The following proposition is an improvement of Theorem 4.1.7. Proposition 7.3.14. Let µ be a normal fuzzy subgroup of a group G such that G/µ∗ is of square free order. Let x, y ∈ G. Then the following assertions hold. (1) If F Oµ (x) divides F Oµ (y), then µ(y) ≤ µ(x). (2) If F Oµ (x) = F Oµ (y), then µ(y) = µ(x), i.e., µ has property (∗). Proof. Let g be the natural homomorphism of G onto G/µ∗ . Then F Oµ (x) = o(g(x)) and µ(x) = (g(µ))(g(x)) for all x ∈ G. However, g(µ) is normal in G/µ∗ by Theorem 1.3.13 since µ is normal in G. Thus the desired conclusions follow by Theorem 4.1.7.

7.4 Cyclic Fuzzy Subgroups and Cyclic Fuzzy p-subgroups In this section, we introduce the notion of a fuzzy cyclic subgroup and a fuzzy cyclic p-subgroup of a group. We use these notions to obtain characterizations for a ﬁnite cyclic group and for a ﬁnite cyclic p-group. We also obtain another characterization for a ﬁnite cyclic p-group using the notion of the order of a fuzzy subgroup introduced in [17] and presented in Chapter 1. Finally, we use the notion of the fuzzy direct product to obtain necessary and suﬃcient conditions for a ﬁnite group to be the direct product of its cyclic p-subgroups or of its cyclic subgroups. The results here mainly from [4]. For a fuzzy subgroup µ of a group G, recall that µ∗ = {x ∈ G | µ(x) = µ(e)} is a subgroup of G. If µ∗ = {e}, then F Oµ (x) = o(x) for all x in G. Recall also that in Section 7.1, the relation ∼ on the set of all fuzzy subgroups of a group G was deﬁned as follows: ∀ µ, ν ∈ F(G), µ ∼ ν if and only if µ and ν have the same chain of level subgroups. Let µ be a fuzzy subgroup of a group G. Recall that if there exists a positive integer n such that for all x ∈ G, µ(xn ) = µ(e), then the least such positive integer is called the order of µ, written O(µ). If no such n exists, µ is said to have inﬁnite order. Let µ be a fuzzy subgroup of a group G. Let p be a prime. Recall that µ is said to be a fuzzy p-subgroup of G if F Oµ (x) is a power of p for all x in

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G. We also recall that the order of a fuzzy p-subgroup is a power of p if the group is ﬁnite. The next result follows from Section 7.1. Theorem 7.4.1. Suppose that G is a ﬁnite group. Then G is the direct product of its normal subgroups G1 , ..., Gn if and only if there exist fuzzy subgroups µ1 , ..., µn of maximal chain of G1 , ..., Gn , respectively, such that their fuzzy direct product is a fuzzy subgroup of G. We introduce the notion of a fuzzy cyclic subgroup of a group G and obtain a characterization for ﬁnite cyclic groups. Deﬁnition 7.4.2. A fuzzy subgroup µ of a group G is said to be a fuzzy cyclic subgroup of G if for all t ∈ Im(µ), there exists element z ∈ µt such that ν(z) ≤ ν(x) for all fuzzy subgroups ν of G and for all x in µt . For an example, consider the Klein 4-group G = {e, a, b, c} and consider the fuzzy subgroups µ, ν and ρ of G deﬁned by µ(e) = r2 , µ(a) = r1 , µ(b) = µ(ab) = r0 , ν(e) = s2 , ν(b) = s1 , ν(a) = ν(ab) = s0 and ρ(e) = t2 , ρ(ab) = t1 , ρ(a) = ρ(b) = t0 , where r0 , r1 , r2 , s0 , s1 , s2 , and t0 , t1 , t2 are elements of [0,1] such that r0 < r1 < r2 , s0 < s1 < s2 , and t0 < t1 < t2 . Then clearly G = µr0 , r0 = µ(ab) = µ(b) < µ(a) while ν(b) > ν(a) and ρ(ab) > ρ(a). Thus µ is not fuzzy cyclic. Theorem 7.4.3. Let G be a ﬁnite group. Then G is cyclic if and only if every fuzzy subgroup of G is fuzzy cyclic. Proof. Suppose every fuzzy subgroup of G is fuzzy cyclic. Let µ be a fuzzy subgroup of G with Im(µ) = {t0 , ..., tn }, t0 < t1 < ... < tn . Let z be an element of G = µt0 such that ν(z) ≤ ν(x) for all x ∈ G and for all fuzzy subgroups ν of G. Suppose G = z. Let x ∈ G\z. Then χz is a fuzzy subgroup of G and χz (z) > χz (x), a contradiction. Thus G = z. Conversely, suppose G is cyclic. Then G = z for some z ∈ G. Let µ be a fuzzy subgroup of G. Suppose t ∈ Im(µ). Then µt is cyclic and so µt = z k for some k ∈ N. Let ν be a fuzzy subgroup of G. Let x ∈ µt . Then x = z km for some m ∈ Z. Thus ν(x) = ν(z km ) ≥ ν(z k ) ∧ ... ∧ ν(z k ) = ν(z k ). Hence µ is fuzzy cyclic. Theorem 7.4.4. Let G be a ﬁnite group and µ be a fuzzy subgroup of G. Then the following conditions are equivalent. (1) µ is fuzzy cyclic and µ∗ = {e}. (2) F Oµ (z) = o(G) for some z in G. (3) G is Abelian and O(µ) = o(G). Proof. (1) ⇒ (2) Since µ is fuzzy cyclic, an argument as in the proof of the last theorem shows that G is cyclic. Suppose that G = z for some z in G.

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Then the condition that µ∗ = {e} is equivalent to F Oµ (x) = o(x) for all x in G. Hence F Oµ (z) = o(z) = o(G). (2) ⇒ (1) Suppose that F Oµ (z) = o(G) for some z in G. Since F Oµ (z) divides o(z), o(z) = o(G). Thus G = z. Hence by Theorem 7.4.3, every fuzzy subgroup of G is fuzzy cyclic and so µ is fuzzy cyclic. Furthermore, if o(µ∗ ) = n and m = o(G)/n, then µ∗ = z m and hence F Oµ (z) = m. However, we have F Oµ (z) = o(G) and so µ∗ = {e}. (2) ⇒ (3) Suppose F Oµ (z) = o(G) for some z in G. Then from the proof of (1) ⇔ (2), it follows that G is cyclic and hence Abelian. This together with Proposition 1.6.5 and Corollary 2.3.20 (2) imply that O(µ) = F Oµ (z). Thus O(µ) = o(G). (3) ⇒ (2) Suppose that G is Abelian and O(µ) = o(G). Then by Corollary 2.3.20 (2), there exists z in G such that F Oµ (z) = O(µ). Hence F Oµ (z) = o(G). We next introduce the notion of a fuzzy cyclic p-subgroup of a group G and use it to obtain a characterization of ﬁnite cyclic p-groups. Deﬁnition 7.4.5. A fuzzy subgroup µ of a group G is said to be a fuzzy cyclic p-subgroup of G if for all x, y ∈ G, (1) µ(x) = µ(y) ⇔ x = y, (2) µ(x) < µ(y) ⇔ x ⊃ y. It follows easily that the condition x = y(x ⊃ y) implies ν(x) = ν(y) (ν(x) ≤ ν(y)) for all fuzzy subgroups ν of G and for all x, y ∈ G. It follows from Theorem 7.4.8 that if G is ﬁnite and µ satisﬁes the conditions in the previous deﬁnition, then µ is fuzzy cyclic and there exists a ﬁxed prime number p such that O(µ) is a power of p. This is the reason µ is called a fuzzy cyclic p-subgroup of G. For an example, consider a cyclic group G = z of order 6 and the fuzzy subgroup µ of G deﬁned by µ(e) = t2 , µ(z 2 ) = µ(z 4 ) = t1 , µ(z) = µ(z 3 ) = µ(z 5 ) = t0 , where t0 , t1 , t2 are elements of [0, 1] such that t0 < t1 < t2 . Then clearly µ is a fuzzy cyclic group of maximal chain and µ(z) = µ(z 3 ). However z = z 3 and thus µ is not a fuzzy cyclic p-subgroup of G. Theorem 7.4.6. Let G be a ﬁnite group. Then G is a cyclic p-subgroup if and only if G has a fuzzy cyclic p-subgroup. Proof. Suppose G is a cyclic p-group. Then |G| = pn for some n ∈ N. Consider the following chain of subgroups of G, G = G0 ⊃ G1 ⊃ ... ⊃ Gn = {e},

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where Gi is the subgroup of G generated by an element of order pn−i , i = 0, 1, ..., n. Let µ be the fuzzy subgroup of G which has the above chain as its chain of level subgroups and Im(µ) = {t0 , ..., tn } such that t0 < t1 < ... < tn . If µ(x) = µ(y) (µ(x) < µ(y)) for x, y in G, then x = µti and y = µtj for some i, j ∈ {0, 1, · · · , n}. It follows easily that µ(x) = ti and µ(y) = tj . This together with µ(x) = µ(y) (µ(x) < µ(y)) yields i = j(i < j) and consequently that x = y (x ⊃ y). Hence µ is a fuzzy cyclic p-subgroup of G. Conversely, let µ be a fuzzy cyclic p-subgroup of G with Im(µ) = {t0 , t1 , ..., tn } Im(µ) = {t0 , t1 , ..., tn } such that t0 < t1 < ... < tn . Let z ∈ G be such that µ(z) = t0 . Let x ∈ G. Then µ(x) ≥ µ(z) and so z ⊇ x. Hence x ∈ z and so G = z. Let H = h and K = k be subgroups of G. Then either µ(h) ≤ µ(k) or µ(h) ≤ µ(k). Thus we have either H ⊇ K or H ⊆ K. Hence either o(K) divides o(H) or o(H) divides o(K). Thus G is a cyclic p-subgroup. Corollary 7.4.7. Let G be a ﬁnite p-group. Then G is cyclic if and only if G has a fuzzy cyclic p-subgroup. The following result gives a characterization for a fuzzy cyclic p-subgroup of a group. Theorem 7.4.8. Let µ be a fuzzy subgroup of a ﬁnite group G. Then the following conditions are equivalent. (1) µ is a fuzzy cyclic p-subgroup. (2) All subgroups of G are chained as level subgroups of µ. (3) µ is a fuzzy cyclic and a fuzzy p-subgroup of maximal chain. Proof. (1) ⇒ (2) Let µ be a fuzzy cyclic p-subgroup of G. Then G is a cyclic p-group by Theorem 7.4.6. Let H be a subgroup of G. Then H = h for some h ∈ H. Now µµ(h) = h , where h is an element of G. Clearly, µ(h ) = µ(h). However, since µ is a fuzzy cyclic p-subgroup, we have that h = h and hence µµ(h) = H. Therefore, all subgroups of G are chained as level subgroups of µ. (2) ⇒ (1) Suppose G = G0 ⊃ G1 ⊃ ... ⊃ Gn = {e} is the chain of all subgroups of G and are chained as level subgroups of µ. If / G1 , we have x ∈ G0 /G1 , then x = Gi for some i ∈ {0, 1, · · · , n}. Since x ∈ G = x. Also it is clear that G is a cyclic p-group. By a similar technique that was used in the ﬁrst part of the proof of Theorem 7.4.6, it follows that µ is a fuzzy cyclic p-subgroup of G. (2) ⇒ (3) Let µ be a fuzzy cyclic p-subgroup of G. By Theorem 7.4.6, G is a cyclic p-group. Since G is a cyclic p-group, µ is fuzzy cyclic by Theorem

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7.4.3 and µ is a fuzzy p-subgroup by Theorem 2.3.17. Also, as in the proof of (1) ⇔ (2), it follows from condition (2) that µ is of maximal chain. (3) ⇒ (2) Since µ is of maximal chain, µ∗ is trivial and hence G is a pgroup by Proposition 7.2.2. Also, since µ is fuzzy cyclic, it follows as in the proof of Theorem 7.4.3 that G is cyclic. Thus since G is a cyclic p-group, G has a unique maximal series which is the chain of all its subgroups. However, since µ is of maximal chain, it has the unique maximal series of G as its chain of level subgroups. By the previous theorem, we have the following corollary which gives another characterization of a ﬁnite cyclic p-group. Corollary 7.4.9. Let G be a ﬁnite group. Then the following properties are equivalent. (1) G is a cyclic p-group. (2) All fuzzy subgroups of maximal chain are equivalent. (3) Every fuzzy subgroup of maximal chain is a fuzzy cyclic p-subgroup. Combining the above theorem with Theorem 7.4.4, we obtain the following result. Corollary 7.4.10. Let G be a ﬁnite p-group and µ a fuzzy subgroup of G. Then the following properties are equivalent. (1) µ is a fuzzy cyclic p-subgroup. (2) µ is fuzzy cyclic and of maximal chain. (3) µ is of maximal chain and F Oµ (z) = o(G) for some z in G. (4) G is abelian, O(µ) = o(G), and µ is of maximal chain. Remark 7.4.11. It follows easily that the equivalence relation ∼ preserves the following properties concerning fuzzy subgroups µ and ν of G : (1) Fuzzy order of an element of a group G with respect to a fuzzy subgroup µ, (that is, µ ∼ ν ⇒ F Oµ (x) = F Oν (x) for all x in G) and hence it preserves the order of a fuzzy subgroup, (that is, µ ∼ ν ⇒ O(µ) = O(ν)); (2) Fuzzy p-subgroup (that is, µ ∼ ν and µ is fuzzy p-subgroup ⇒ ν is fuzzy p-subgroup); (3) Fuzzy cyclic subgroup; (4) Fuzzy cyclic p-subgroup. We now consider direct products. The following result is a direct consequence of Theorems 7.4.3 and 7.4.1. Theorem 7.4.12. Let G be a ﬁnite group and G1 , ..., Gn be cyclic subgroups of G, where n ∈ N. Then G is direct product of G1 , ..., Gn if and only if there exist fuzzy cyclic subgroups of G1 , ..., Gn of maximal chain such that their fuzzy direct product is a fuzzy subgroup of G.

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Theorem 7.4.13. Let G be a ﬁnite group and let p1 , ..., pn be the prime factors of o(G). Then G is cyclic if and only if there exist fuzzy cyclic pi -subgroups of pi -primary components of G, i = 1, ..., n, such that their fuzzy direct product is a fuzzy subgroup of G. ⊗ Proof. Suppose G is cyclic. Let I = {1, ..., n}. Suppose G = i∈I Gi , where the Gi are the primary pi -components of G. Then by Theorem 7.4.1, there exist fuzzy subgroups µi of Gi of maximal chain such that their direct product is a fuzzy subgroup of G, i = 1, ..., n. Since Gi is a cyclic pi -subgroup of G, it follows from Corollary 7.4.10 that µi is a fuzzy cyclic pi -subgroup of Gi for all i = 1, ..., n. ⊗ Conversely, let µ = i∈I µi be a fuzzy subgroup of G, where the µi are fuzzy cyclic pi -subgroups of pi -primary components of G. By Theorem 7.4.6, it follows that the Gi are cyclic pi -subgroups of G. By Theorem 7.4.1, we have that G is the direct product of cyclic pi -subgroups of G. Since the pi are distinct, G is cyclic. The next result follows from Theorem 7.4.13, Corollary 7.4.10 and Theorem 7.1.16. Corollary 7.4.14. Let G be a ﬁnite Abelian group. Then a fuzzy subgroup of G is fuzzy cyclic of maximal chain if and only if it is a fuzzy direct product of fuzzy cyclic pi -subgroups of the pi -primary components of G. ⊗Let I = {1, 2, ..., n}. Suppose that G is a ﬁnite cyclic group and G = i∈I Gi , where the Gi are the pi -primary components of G and p1 , ..., pn are the prime factors of o(G). Let µ, µ1 , ..., µn be fuzzy subgroup of G, G1 , ..., Gn , respectively, such that µ is the fuzzy direct product of the µi . Then µ1 , ..., µn are called fuzzy pi -primary components of µ. For such fuzzy subgroups µ of G, by preserving order, we mean, ﬁrst ﬁx the ordering of p1 , ..., pn and then that of the fuzzy pi -primary components of µ. If PIµ and PIν ∈ A for I = {1, ..., n}, we say that µ is associated to ν if whenever µi is equal (equivalent) to νi ∀i = 1, ..., n, then µ is equal (equivalent to ν. Theorem 7.4.15. Let G be a ﬁnite cyclic group. Then all associated fuzzy subgroups of G which are the fuzzy direct product of their fuzzy primary components and preserving order are equivalent. ⊗ Proof. Let G = i∈I Gi , where the Gi are the pi -primary components of G and I = {1, 2, ..., n}. Then it is known that this decomposition is unique. Suppose A is the set of all PIµ , where µ is a fuzzy subgroup of G and the µi are the fuzzy pi -primary components of µ. Suppose that PIµ , PIν are elements of A. Since the µi , νi are fuzzy cyclic pi -subgroups, they are of maximal chain by Theorem 7.4.8. Thus by Corollary 7.4.9, we have that µi is equivalent to νi for all i = 1, ..., n . By Theorem 7.1.7, it follows that µ is equivalent to ν.

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7.5 Fuzzy p∗-subgroups In this section, we introduce the notion of a fuzzy p∗ -subgroup and characterize fuzzy subgroups of torsion groups and cyclic groups by their minimal fuzzy p-subgroups and minimal fuzzy p∗ -subgroups. We extend the notion of the direct product of ﬁnitely many fuzzy subgroups to the direct sum of an arbitrary family of fuzzy subgroups of a commutative group. We also give a condition for a fuzzy subgroup of the direct sum of groups to be the direct sum of its projections. We show that a fuzzy subgroup of a torsion group is completely characterized by its minimal fuzzy p-subgroups. We generalize the notion of a fuzzy p-subgroup by presenting the notion of a fuzzy p∗ -subgroup. We show that a fuzzy subgroup of a cyclic group is completely characterized by its minimal fuzzy p∗ -subgroups. The results of this section are mainly from [18]. Let f be a homomorphism of a group G into a group H. Let µ be a fuzzy subgroup of G. Recall that µ is called f -invariant if f (x) = f (y) implies µ(x) = µ(y) for all x, y ∈ G. Lemma 7.5.1. Let f be a homomorphism of a group G onto a group H. Let µ and ν be fuzzy subgroups of G and H, respectively. Then the following assertions hold. (1) If F Oµ (x) < ∞ for some x ∈ G, then F Of (µ) (f (x)) < ∞ and F Of (µ) (f (x)) divides F Oµ (x). (2) If µ is f -invariant, then F Of (µ) (f (x)) = F Oµ (x) for all x ∈ G. (3) F Oν (f (x)) = F Of −1 (ν) (x) for all x ∈ G. Proof. (1) Let F Oµ (x) = n. Then (f (µ))(f (x)n ) ≥ µ(xn ) = µ(e) and so F Of (µ) (f (x)) divides F Oµ (x) by Proposition 1.6.5. (2) For every integer n, (f (µ))(f (x)n ) = (f (µ))(f (xn )) = ∨{µ(z) | f (z) = f (xn ), z ∈ G} = µ(xn ). (3) For every integer n, (f −1 (ν))(xn ) = ν(f (xn )) = ν((f (x))n ). Proposition 7.5.2. Let f be a homomorphism of a group G onto a group H and let µ and ν be fuzzy subgroups of G and H, respectively. Then the following assertions hold. (1) If O(µ) < ∞, then O(f (µ)) < ∞ and O(f (µ)) divides O(µ). (2) If µ is f -invariant, then O(f (µ)) = O(µ). (3) O(f −1 (ν)) = O(ν). Proof. The proof follows easily from Lemma 7.5.1.

The following result now follows. Proposition 7.5.3. Let µ be a fuzzy subgroup of a group G such that µ∗ a normal subgroup of G. Then µ is a fuzzy p-group if and only if G/µ∗ is a p-group.

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Note that the order of a fuzzy p-subgroup of G is a power of p if this order is ﬁnite. Also, the homomorphic image and the homomorphic preimage of a fuzzy p-subgroup are again fuzzy p-subgroups. Relevant classical group-theoretical results and deﬁnitions can be found in [30]. Deﬁnition 7.5.4. Let µ be a fuzzy subgroup of an Abelian group G. Then µ is called fuzzy torsion if F Oµ (x) is ﬁnite for all x ∈ G. Proposition 7.5.5. Let µ be a fuzzy subgroup of an Abelian group G. Then µ is fuzzy torsion if and only if G/µ∗ is torsion. We denote the minimal fuzzy p-subgroup of G containing the fuzzy subgroup µ of G by µ(p) when it exists. That µ(p) does not exist in general was shown in Section 7.3. The following result is immediate from Corollary 7.2.12. However the converse does not hold as is shown in the next example. Proposition 7.5.6. Every fuzzy torsion subgroup of an Abelian group can be expressed as the intersection of the µ(p) . Example 7.5.7. Let p1 < p2 < ... denote the primes. Deﬁne a fuzzy subgroup µ of the additive group Z of all integers as follows: ∀x ∈ Z,  if x = 0, 1 / p1 ...pi+1 Z for some i ≥ 1, µ(x) = 1 − 1/(i + 1) if x ∈ p1 ...pi Z and x ∈  0 otherwise. For each prime pi , deﬁne the fuzzy subgroup µi of Z as follows: ∀x ∈ Z, 1 if x ∈ pi Z, µi (x) = 1 − 1/i otherwise. Let x ∈ G. Then / pi+1 Z µ(x) = 1 − 1/(i + 1) ⇔ x ∈ p1 Z ∩ ... ∩ pi Z, x ∈ ⇔ (∩µi )(x) = 1 ∧ ... ∧ 1 ∧ (1 − 1/(i + 1) ∧ {tj | j = i + 1, i + 2, ...} = 1 − 1/(i + 1), where tj is either 1 or 1 − 1/(j + 1) for j = i + 1, i + 2, .... Thus it follows that µ = ∩µi . Also, µi = µ(pi ) for each i, but F Oµ (1) is inﬁnite. Deﬁnition 7.5.8. Let Gi be an Abelian group and let µi be a fuzzy subgroup of Gi ∀ i ∈ I. The direct sum ⊕i∈I µi is the fuzzy subgroup of the direct sum ⊕i∈I Gi deﬁned by ⊕i∈I µi ( xi ) = ∧{µi (xi ) | i ∈ I}.

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Theorem 7.5.9. Let µ be a fuzzy subgroup of the direct sum of the Abelian groups Gi , where i ∈ I. Let µi be the projection of µ on Gi for all i ∈ I. Suppose (F Oµj (x), F Oµk (y)) = 1 for all j, k ∈ I such that j = k and for all x ∈ Gj and y ∈ Gk . Then µ = ⊕i∈I µi . Proof. Since all but ﬁnitely many coordinates of each of the elements of ⊕i∈I Gi are the identity, the proof is same as in the case of a ﬁnite direct product, Theorem 7.2.15. Corollary 7.5.10. Let µ be a fuzzy subgroup of a torsion group G. Then µ = ⊕µp , where µp is the projection of µ on the p-primary subgroup Gp of G for each prime p. Proof. The proof follows by Proposition 1.6.5 and Theorem 7.5.9.

Lemma 7.5.11. Let µ be a fuzzy subgroup of a torsion group G. For each prime p, the projection of µ(p) on the p-primary subgroup Gp of G is equal to the projection µp of µ on Gp . Furthermore, the projection of µ(p) on the p-complement subgroup Gp of G has the constant image µ(0). In other words, µ(p) may be identiﬁed with the projection of µp on G. Proof. Let x ∈ Gp . Then o(x) is a power of p and so µ(q) (x) = µ(0) for all primes q = p by Proposition 1.6.5. Thus µ(p) (x) = µ(x) by Proposition 7.5.6. On the other hand, if x ∈ Gp , then (o(x), p) = 1 and so µ(p) (x) = µ(0) by Proposition 1.6.5. By Lemma 7.5.11, Corollary 7.5.10 can also be derived from Proposition 7.5.6. Let µ and ν be fuzzy subgroups of the groups G and H, respectively. Recall that µ and ν are said to be isomorphic if there exists a isomorphism f of G onto H such that f (µ) = ν. Theorem 7.5.12. Let µ and ν be fuzzy subgroups of the torsion groups G and H, respectively. Then µ and ν are isomorphic if and only if µ(p) and ν(p) are isomorphic for every prime p. Proof. By Proposition 7.5.6, µ(p) and ν(p) exist for every prime p. The necessity follows by Proposition 7.5.2. We now consider the suﬃciency. For each prime p, let fp : G → H be the isomorphism such that fp (µ(p) ) = ν(p) . Since G and H are torsion, gp = fp |Gp is an isomorphism from Gp onto Hp and gp (µp ) = νp by Lemma 7.5.11. Deﬁne f : G → H by f ( xp ) = (gp (xp )), where xp belongs to the component Gp of G. Then f is clearly an isomorphism, and f (µ) = ν by Corollary 7.5.10. Due to Proposition 7.5.6 and Theorem 7.5.12, the study of fuzzy subgroups of a torsion group is reduced to the study of its fuzzy p-subgroups. The following example shows that Theorem 7.5.12 does not hold if G and H are not torsion groups.

7.5 Fuzzy p∗ -subgroups

195

Example 7.5.13. For all n ∈ N, deﬁne the fuzzy subgroup ξn of Z by for all k ∈ Z, 1 if k ∈ nZ, ξn (k) = 0 otherwise, and let µ = ξ5 ⊕ ξ6 and ν = ξ3 ⊕ ξ10 . Then µ and ν are fuzzy torsion, and µ(p) and ν(p) are isomorphic for every prime p. However, µ and ν are not isomorphic. We now introduce the notion of a fuzzy p∗ -subgroup. Deﬁnition 7.5.14. Let µ be a fuzzy subgroup of a group G and p a prime. Then µ is said to be a fuzzy p∗ -subgroup if for all x ∈ G, ∧{n ∈ N | µ(x) < µ(xn )} is a power of p whenever this minimum exists. Lemma 7.5.15. Let ν be a fuzzy p∗ -subgroup of a group G and let n, m, t be t positive integers such that n = pt m and (p, m) = 1. Then ν(xn ) = ν(xp ) for all x ∈ G. t

t

Proof. Let x ∈ G. Since n = pt m, ν(xp ) ≤ ν(xn ). Assume that ν(xp ) < t ν(xn ). Set y = xp . Then ν(y) < ν(y m ). Since ν is a fuzzy p∗ -subgroup, there exists s ∈ N such that ps = ∧{k ∈ N | ν(y) < ν(y k )} ≤ m. Thus ν(y j ) = ν(y) for j = 1, ..., ps − 1. By the Euclidean algorithm, there are integers q and r such that m = ps q + r and 0 ≤ r < ps . Now r = 0 since (p, m) = 1. Thus s s s ν(y) < ν(y p ) ≤ ν(y p q ) and ν(y) = ν(y r ). Since ν(y p q ) > ν(y) = ν(y r ), s s ν(y m ) = ν(y p q y r ) = ν(y p q ) ∧ ν(y r ) = ν(y r ) = ν(y) by comments following t Lemma 1.2.5. However, this is impossible. Thus ν(xp ) = ν(xn ). Proposition 7.5.16. Let µ be a fuzzy subgroup of a group G. Let p be a prime. Then there exists a unique minimal fuzzy p∗ -subgroup, written µ(p)∗ , of G containing µ. Proof. Let I = {νi | i ∈ I} be the set of all fuzzy p∗ -subgroups of G containing µ. Then I is nonempty since a constant fuzzy subgroup is a fuzzy p∗ -subgroup. Set ν = ∩i∈I νi . For x ∈ G, let ν(x) < ν(xn ) for some positive integer n, where t n = pt m and (p, m) = 1. Then for all i ∈ I, νi (xn ) = νi (xp ) since νi is a t fuzzy p∗ -subgroup. Hence ν(xn ) = ν(xp ). Thus ν is the desired µ(p)∗ . Lemma 7.5.17. Let µ be a fuzzy subgroup of a group G and let F Oµ (x) = n, where x ∈ G and n ∈ N. If m is an integer such that (n, m) = 1, then µ(xm ) = µ(x). Proof. Since (n, m) = 1, there exist integers s and t such that ns + mt = 1. Thus µ(x) = µ(xns+mt ) = µ((xn )s (xm )t ) ≥ µ(xn ) ∧ µ(xm ) = µ(xm ) ≥ µ(x). Theorem 7.5.18. Every fuzzy p-subgroup of a group G is a fuzzy p∗ -subgroup of G.

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Proof. Let x ∈ G and let n = pt m be a positive integer such that µ(x) < t µ(xn ) and (p, m) = 1. Set y = xp . Then F Oµ (y) is a power of p. Hence t (F Oµ (y), m) = 1. Thus µ(xp ) = µ(y) = µ(y m ) = µ(xn ) by Lemma 7.5.17. Thus the notion of fuzzy p∗ -subgroups is an extension of the notion of fuzzy p-subgroups. Let p be a prime p. Then µ(p) does not exist in general, but µ(p)∗ always exists. Example 7.5.19. Deﬁne the fuzzy subset µ of Z as follows: ∀n ∈ Z, 1 if n = 0, µ(n) = 0 otherwise. Then µ is not a fuzzy p-subgroup of Z, but µ is vacuously a fuzzy p∗ -subgroup of Z. Proposition 7.5.20. Let f be a homomorphism of G onto H. Let µ and ν be fuzzy subgroups of G and H, respectively. Then the following assertions hold. (1) Suppose that µ is f -invariant. If µ is a fuzzy p∗ -subgroup of G, then f (µ) is a fuzzy p∗ -subgroup of H. (2) If ν is a fuzzy p∗ -subgroup of H, then f −1 (ν) is a fuzzy p∗ -subgroup of G. Proof. (1) Let n ∈ N and y ∈ H. Then (f (µ))(y n ) = ∨{µ(z) | f (z) = y n , z ∈ G} = µ(z)∀z ∈ G such that f (z) = y n . Hence (f (µ))(y n ) = µ(wn ), where w ∈ G is such that f (w) = y. (2) Let n ∈ N and x ∈ G. Then (f −1 (ν))(xn ) = ν(f (xn )) = ν(f (x)n ), where y ∈ H and x ∈ G. Proposition 7.5.20(1) does not hold in general if µ is not f -invariant. This can be seen from the following example. Example 7.5.21. Deﬁne the fuzzy subset µ of Z as follows:∀n ∈ Z,   1 if n = 0, µ(n) = 12 if n ∈ 3Z\{0},  0 otherwise. Let f be a homomorphism of Z onto Z2 . Then µ is a fuzzy 3∗ -subgroup, but f (µ) is a fuzzy 2∗ -subgroup. Note that f (1) = f (3) while µ(1) = µ(3) and so µ is not f -invariant. We now consider fuzzy subgroups of a cyclic group. Corresponding to a fuzzy subgroup µ of a cyclic group x, there exists a (ﬁnite or inﬁnite) strictly increasing sequence {ni } of positive integers as follows:

7.5 Fuzzy p∗ -subgroups

197

n1 = ∧{k ∈ N | µ(xk ) = ∧{µ(y) | y ∈ x}} = 1, n2 = ∧{k ∈ N | µ(xk ) = ∧({µ(y) | y ∈ x}\ {µ(x)})}, n3 = ∧{k ∈ N | µ(xk ) = ∧({µ(y) | y ∈ x}\ {µ(x), µ(xn2 )})}, and so on. The sequence {ni } is independent of any speciﬁc choice of a generator of the cyclic group x. In fact, if y is another generator of x , we have µ(xr ) = µ(y r ) for all integers r. Also, it follows that ni |ni+1 for all i = 1, 2, .... Let p be a prime p. Let{mi } be a sequence of nonnegative integers such that pmi |ni and pmi +1 ni , i = 1, 2, .... Consider the strictly increasing maximal subsequence {mij } of {mi } such that each index ij is minimal. The sequence {mi } depends on the particular choice of a prime p. Theorem 7.5.22. Let µ be a fuzzy subgroup of a cyclic group x. Let ni and mi be deﬁned as above. Let p be a prime. Then µ(p)∗ is determined as follows: (1) If F Oµ (x) is ﬁnite, then  n m m  µ(x ij−1 ) if t ∈ (p ij−1 )Z\(p ij )Z, m µ(p)∗ (xt ) = µ(e) otherwise, i.e., t ∈ (p ij Z)and mij is the last element of  the subsequence, (2) If F Oµ (x) is inﬁnite, then  µ(xnij−1 ) if t ∈ (pmij−1 )Z\(pmij )Z,    ∨{µ(y)|y ∈ x, y = e} if t ∈ (pmij )Z\{0}and mij is the last µ(p)∗ (xt ) = element of the subsequence,    µ(e) if t = 0. Proof. (1) Since µ(p)∗ is a fuzzy p∗ -subgroup, the µ(p)∗ (xt ) are equal for all ni t ∈ (pmij−1 )Z\(pmij )Z. Also, µ(x j −1 ) is the maximal value of the µ(xt ) ni for these t. However, µ(p)∗ contains µ. Thus µ(p)∗ (xt ) = µ(x j −1 ) for all mij such t since µ(p)∗ is minimal. Next, let t ∈ (p )Z, where mij is the last element of the subsequence. If (F Oµ (x), p) = 1, then F Oµ (x) ∈ (pmij )Z and similarly µ(p)∗ (xt ) = µ(e). If (F Oµ (x) , p) = 1, then mij = 1 and we have µ(p)∗ (x) = µ(p)∗ (xF Oµ (x) ) = µ(e). (2) The ﬁrst part, even if it involves inﬁnitely many steps, is quite similar to the ﬁrst part of (1). For the second part, the µ(p)∗ (xt ) are equal for all t ∈ (pmij )Z\{0} since mij is the last element of the subsequence and µ(p)∗ is a fuzzy p∗ -subgroup. Thus for all such t, µ(p)∗ (xt ) ≥ ∨{µ(xt ) | t ∈ (pmij )Z\{0}} = ∨{µ(y) | y ∈ x, y = e} since µ(p)∗ contains µ. However, µ(p)∗ is minimal. Hence µ(p)∗ (xt ) = ∨{µ(y) | y ∈ x, y = e} for all t ∈ (pmij )Z\{0}. Corollary 7.5.23. Assume the hypothesis of Theorem 7.5.22. If F Oµ (x) is e} = µ(e), ﬁnite, then µ(p)∗ = µ(p) . If F Oµ (x) is inﬁnite, ∨{µ(y) | y ∈ x, y =

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and if {mij } is a ﬁnite sequence, then µ(p)∗ = µ(p) . Otherwise, µ(p) does not exist. Corollary 7.5.24. Let µ be a fuzzy subgroup of a cyclic group G = x . Then µ is a fuzzy p∗ -subgroup if and only if either G/µ∗ is a p-group or given n t ∈ [0, µ (e)], the t-level subgroup of µ is xp for some nonnegative integer n. Proof. The proof follows by applying Proposition 7.5.3 to Corollary 7.5.23. Let p be a prime. Then Theorem 7.5.22 and Corollary 7.5.23 characterize µ(p) and µ(p)∗ for a fuzzy subgroup µ of a cyclic group. On the other hand, the following theorem and corollary show that a fuzzy subgroup of a cyclic group is completely determined by its fuzzy p∗ -subgroups. Theorem 7.5.25. Let µ be a fuzzy subgroup of cyclic group x. Then µ = ∩µ(pi )∗ , where the intersection is over the set of all distinct primes pi . Proof. Since µ(pi )∗ contains µ for each pi , it suﬃces to show that µ ⊃ ∩µ(pi )∗ . Assume the hypothesis of Theorem 7.5.22. Let t ∈ N. Then there exists nk ∈ {ni } such that µ(xnk ) = µ(xt ). If nk is the last element of the sequence {ni }, then µ(xnk ) = ∨ {µ(xs )|s ∈ N} . Let p be a prime such that p is not a factor of nk . Then the subsequence {mij }, which depends on the pi , is precisely {0}. Thus µ(p)∗ (xt ) = µ(xnk ) = µ(xt ) by Theorem 7.5.22. On the other hand, if nk is not the last element of the sequence {ni }, then there exists a prime p such that pmk < pmk+1 and pmk+1 t since nk+1 is a proper multiple of nk , where {mi } is the sequence which depends on the p. However, pmk |t since nk |t. Hence µ(p)∗ (xt ) = µ(xnk ) = µ(xt ) by Theorem 7.5.22. Thus there exists a prime p such that µ(p)∗ (xt ) = µ(xt ) ∀t ∈ N. Therefore, µ ⊃ ∩µ(pi )∗ . Corollary 7.5.26. Let µ and ν be fuzzy subgroups of the cyclic groups G and H, respectively. Then µ and ν are isomorphic if and only if µ(p)∗ and ν(p)∗ are isomorphic for every prime p. Proof. If |G| = |H| is ﬁnite, then the proof follows by Theorem 7.5.12 and Corollary 7.5.23. Thus let G and H be inﬁnite cyclic groups. The necessity is easily obtained by Proposition 7.5.20. We now consider the suﬃciency. For each prime p, let fp : G → H be an isomorphism such that fp (µ(p)∗ ) = ν(p)∗ . Since G and H are inﬁnite cyclic groups, they are isomorphic under exactly two isomorphisms g and h, say, such that g(x) = y and h(x) = y −1 , where x and y are generators of G and H, respectively. Let f be one of these isomorphisms. Then f (µ) = ν by Theorem 7.5.25.

References

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References 1. S. Abou-Zaid, On generalized characteristic fuzzy subgroups of a ﬁnite group, Fuzzy Sets and Systems 43 (1991) 234-241. 183, 186 2. S. Abou-Zaid, On fuzzy subnormal and pronormal subgroups of ﬁnite group, Fuzzy Sets and Systems 47 (1992) 346-349. 3. M. Akg¨ ul, Some properties of fuzzy groups, J. Math. Anal. Appl. 133 (1988) 93-100. 181, 182 4. Y. Alkhamees, Fuzzy cyclic subgroups and fuzzy cyclic p-subgroups, J. Fuzzy Math. 3 (1995) 911-919. 186 5. Y. Alkhamees, Fuzzy direct product of fuzzy subgroups of subgroups, J. Fuzzy Math. 6 (1998) 307 - 318. 168 6. J. M. Anthony and H. Sherwood, A characterization of fuzzy subgroups, Fuzzy Sets and Systems 7 (1982) 297-305. 7. J. M. Anthony and H. Sherwood, Fuzzy subgroups redeﬁned, J. Math. Anal. Appl. 69 (1979)124-130. 8. M. Asaad, Groups and fuzzy subgroups, Fuzzy Sets and Systems 39 (1991) 323328. 183 9. P. Bhattacharya, Fuzzy subgroups: Some characterizations, J.Math Anal. Appl. 128(1987), 241-252. 10. P.S. Das, Fuzzy groups and level subgroups, J.Math. Anal. Appl. 84(1981) 246269. 11. V. N. Dixit, R.Kumar and N. Ajmal, Level subgroups and union of fuzzy subgroups, Fuzzy Sets and Systems 37 (1990) 359-371. 12. B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967. 13. J. G. Kim, On fuzzy orders of elements of fuzzy subgroups, J. Kyungsung Univ. 13 (1992) 251-257. 14. J. G. Kim, Fuzzy subgroups and minimal fuzzy p-subgroups, J. Fuzzy Math. 2 (1994) 913-921. 15. J. G. Kim, Fuzzy orders relative to fuzzy subgroups, Inform. Sci. 80 (1994) 341-348. 16. J. G. Kim, Fuzzy subgroups having the property (∗), Inform. Sci. 80 (1994) 235-241. 183 17. J. G. Kim, Orders of fuzzy subgroups and fuzzy p-subgroups, Fuzzy Sets and Systems 61(1994), 225-230. 177, 186 18. J. G. Kim and H. D. Kim, A characterization of fuzzy subgroups of some abelian groups, Inform. Sci. 80 (1994) 243-252. 192 19. J. G. Kim and H. D. Kim, Some characterizations of fuzzy subgroups: via fuzzy p∗ -subsets and fuzzy p∗ -subgroups, Fuzzy Sets and Systems 102 (1999) 327-332 20. J. G. Kim, Some characterizations of fuzzy subgroups, Fuzzy Sets and Systems 87 (1997) 243-249. 21. R. Kumar, Fuzzy Sylow subgroups, Fuzzy Sets and Systems 46 (1992) 267-271. 178 22. R. Kumar, Fuzzy subgroups, fuzzy ideals and fuzzy cosets, some properties, Fuzzy Sets and Systems 48(1992)267-274. 23. I. J. Kumar, R.K. Saxena and P. Yadav, Fuzzy normal subgroups and fuzzy quotients, Fuzzy Sets and Systems, 46 (1992) 121-132. 167, 181, 182 24. Wang-jin Liu, Fuzzy invariants subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133-139. 25. D. S. Malik and J. N. Mordeson, Fuzzy subgroups of abelian groups, Chinese J. of Math. 19, 2 (1991) 129-145. 175

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26. N. P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225-239. 27. S. Ray, Isomorphic fuzzy groups, Fuzzy Sets and Systems 50 (1992) 201-207. 28. J. S. Rose, A course on group theory, Cambridge University Press, Cambridge, 1985. 167, 171 29. A. Rosenfeld, Fuzzy groups, J. Math Anal. Appl. 35 (1971) 512-517. 30. J. J. Rotman, An Introduction to the Theory of Groups, 3rd ed., Allyn and Bacon, Boston, MA, 1984. 193 31. H. Sherwood, Products of fuzzy subgroups, Fuzzy Sets and Systems 11 (1983) 79-89. 167, 181, 182 32. B. W. Wetherilt, Semidirect products of fuzzy subgroups, Fuzzy Sets and systems 16 (1985) 237-242. 33. L.A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965) 338-353.

8 Equivalence of Fuzzy Subgroups of Finite Abelian Groups

In this chapter, we determine the number of fuzzy subgroups of certain ﬁnite Abelian groups with respect to a suitable equivalence relation. This is motivated by the realization that in a theoretical study of fuzzy groups, fuzzy subgroups are distinguished by their level sets and not by their images in [0, 1]. We ﬁrst present some results on the equivalence of fuzzy subsets of a given set. The results of this chapter are mainly from [4, 9, 10, 11, 12].

8.1 A Relation on the Set of Fuzzy Subsets of a Set Let X be a set. Deﬁne the relation ∼ on FP(X) as follows: ∀µ, ν ∈ FP(X), µ ∼ v if and only if for all x, y ∈ X, µ(x) > µ(y) if and only if ν(x) > ν(y) and µ(x) = 0 if and only if ν(x) = 0. Then it follows easily that ∼ is an equivalence relation on FP(X). If ∼ is restricted to the crisp subsets of X, C(X) = {f | f : X → {0, 1}}, then ∼ reduces to equality of subsets. The condition µ(x) = 0 if and only if ν(x) = 0 states that the supports of µ and ν are equal. This condition does not follow from the condition that µ(x) > µ(y) if and only if ν(x) > ν(y). Example 8.1.1 illustrates this fact. In this section, we consider ∼ on FP(X), where X is a set. Example 8.1.1. Let G =< a > be a cyclic group of order 2. Deﬁne the fuzzy subsets µ and ν of G as follows : ∀x ∈ G, 1 if x = e, µ(x) = t if x = a; 1 if x = e, ν(x) = 0 if x = a, where 0 < t < 1. Clearly, µ(x) > µ(y) if and only if ν(x) > ν(y) for all x, y ∈ G. However, µ∗ = ν ∗ and therefore µ is not equivalent to ν. Proposition 8.1.2. Let µ and ν be fuzzy subsets of a set X. If µ ∼ ν, then |Im(µ)| = |Im(ν)|. John Mordeson, Kiran R. Bhutani, Azriel Rosenfeld: Fuzzy Group Theory, StudFuzz 182, 201– 238 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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Proof. Deﬁne f from Im(µ) into Im(ν) by f (µ(x)) = ν(x) for all x ∈ X. Then it follows easily that f is well-deﬁned, one-to-one, and onto. Example 8.1.3. Consider the symmetric group on three letters, S3 = {e, a, a2 , b, ab, a2 b}. Deﬁne the fuzzy subsets µ and ν of S3 as follows: ∀x ∈ S3 ,   1 if x = e, 1/2 if x = b, µ(x) =  1/3 otherwise;   1 if x = e, ν(x) = 1/2 if x = ab,  1/3 otherwise. Then Im(µ) = Im(ν) and µ∗ = ν ∗ . However, µ(b) > µ(ab), but ν(b) ≯ ν(ab). Therefore, µ is not equivalent to ν. Proposition 8.1.4. Let µ and ν be fuzzy subsets of X. Suppose that for all t ∈ (0, 1], there exists s ∈ (0, 1] such that µt = νs and for all t ∈ (0, 1], there exists s ∈ (0, 1] such that νt = µs . Then µ ∼ ν. Proof. If µ∗ = ∅, then clearly ν ∗ = ∅. Thus ν(x) = 0 = µ(x) for all x ∈ X. Hence µ ∼ ν trivially. Suppose µ∗ = ∅. Let x ∈ µ∗ . Then x ∈ µt for some t ∈ (0, 1]. By hypothesis, there exists s ∈ (0, 1] such that µt = νs . Hence ν(x) ≥ s > 0. Thus x ∈ ν ∗ . Hence µ∗ ⊆ ν ∗ . Similarly, one can show that ν ∗ ⊆ µ∗ . Therefore, µ∗ = ν ∗ . Now let t = µ(x) > µ(y) for x, y ∈ G. By hypothesis, x ∈ νs = µt for some s ∈ (0, 1]. If ν(x) ≤ ν(y), then ν(y) ≥ s and so y ∈ νs = µt , a contradiction. Hence ν(x) > ν(y). Similarly, ν(x) > ν(y) implies µ(x) > µ(y). We now show a partial converse to the above proposition. By the notation (1, 1], we mean the empty set. Proposition 8.1.5. Suppose µ and ν are fuzzy subsets of X such that µ ∼ ν. Then for all t ∈ [0, 1] , there is an s ∈ [0, 1] such that µt = νs or µt = ν −1 (s, 1]. Proof. Let a = ∨{µ(x) | x ∈ X} and b = ∨{ν(x) | x ∈ X}. For t = a, choose s = b. Then clearly µt = νs . For t < a, consider t1 = ∧{µ(x) | x ∈ µt }. Suppose t1 = µ(y) for some y ∈ X. Then s can be chosen to be ν(y). To show that µt is contained in νs , let x ∈ µt . Then µ(x) ≥ t and so µ(x) ≥ t1 = µ(y). Since µ ∼ ν, ν(x) ≥ ν(y) = s. Thus x ∈ νs . Hence µt ⊆ νs .The reverse inclusion can be shown similarly. Suppose t1 = µ(y) for all y ∈ X. Then µt = µt1 = µ−1 (t1 , 1] = −1 µ (t, 1]. Let s = ∧{ν(x) | x ∈ µt }. Let x ∈ µt . Then µ(x) > t1 and so µ(x) > µ(y) for some y ∈ µt . Since µ ∼ ν, ν(x) > ν(y) ≥ s. Thus x ∈ ν −1 (s, 1]. Hence µt ⊆ ν −1 (s, 1]. Now let x ∈ ν −1 (s, 1]. Then ν(x) > s and so ν(x) > ν(y) for some y ∈ µt . Since µ ∼ ν, µ(x) > µ(y) > t. Hence x ∈ µt . Thus ν −1 (s, 1] ⊆ µt and so µt = ν −1 (s, 1]. Now consider the case where a < 1 and t > a. Then µt = ∅. Suppose b < 1. Then ν1 = ∅ = µt . If for all x ∈ X, ν(x) < b = 1, then

8.1 A Relation on the Set of Fuzzy Subsets of a Set

203

νb = ∅ = µt . If ν(x) = 1 = b for some x ∈ X, then for any s ∈ [0, 1], νs = ∅. In this case, choose s = 1. Then ν −1 (s, 1] = ∅ = µt . The following examples show that the converse of Proposition 8.1.4 does not hold. Example 8.1.6. Let X = {x}. Deﬁne the fuzzy subsets µ and ν of X as follows: µ(x) = 1/2 and ν(x) = 1. Then µ ∼ ν. Let t = 3/4. Then µt = ∅, but νs = X for all s ∈ [0, 1]. However, Proposition 8.1.5 assures that µ3/4 = ν (−1) (1, 1]. Example 8.1.7. Let X = [0, 1]. Deﬁne the fuzzy subset µ of X by µ(x) = 1− 12 x if x ∈ [0, 1) and µ(1) = 0. Deﬁne the fuzzy subset ν of X by ν(x) = 1−x for all x ∈ [0, 1]. Then µ ∼ ν and 1 ∈ Im(µ), 1 ∈ Im(ν). Let t = 1/2. Then µt = [0, 1). However, there is no s ∈ [0, 1] such that νs = [0, 1). Note, µ1/2 = ν (−1) (0, 1]. In the next example, we show that the conditions t > 0 and s > 0 in Proposition 8.1.4 are needed. This example also shows that the converse of Proposition 8.1.5 doesn’t hold. Example 8.1.8. Let X = {x}. Deﬁne the fuzzy subsets µ and ν of X as follows: µ(x) = 1/2 and ν(x) = 0. If t ∈ [0, 1/2], let s = 0. Then µt = X = νs . If t ∈ (1/2, 1], let s ∈ (0, 1].Then µt = ∅ = νs . However, µ and ν are not equivalent. We now consider an equivalence of fuzzy subsets for which the converse of Proposition 8.1.4 holds. Deﬁnition 8.1.9. Let X be a set and µ a fuzzy subset of X. Let x ∈ X. Suppose ∀ > 0, there exists y ∈ X such that µ(x) + > µ(y) > µ(x). Then µ(x) is called a right µ-limit point of Im(µ). Let µR = {x ∈ X | µ(x) is a right µ-limit point of Im(µ)}. Deﬁnition 8.1.10. Let X be a set and let µ and ν be fuzzy subsets of X such that µ ∼ ν. If µR = ν R , then µ and ν are said to be strongly equivalent written µ ≈ ν. It is clear that the notion of strong equivalence is an equivalence relation. Also, if µ is a fuzzy subset of a set X such that Im(µ) is ﬁnite, then µR = ∅. Hence when considering the equivalence of fuzzy subgroups of ﬁnite groups, the notions of equivalence and strong equivalence are the same. The notion of strong equivalence will be used in the consideration of fuzzy subgroups of inﬁnite groups. Theorem 8.1.11. Let µ and ν be fuzzy subsets of a set X. Then µ ≈ ν if and only if µ∗ = ν ∗ and for all t ∈ [0, 1], µt = ∅ implies there is an s ∈ [0, 1] such that µt = νs and for all s ∈ [0, 1], νs = ∅ implies there is a t ∈ [0, 1] such that νs = µt .

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Proof. Suppose that the conditions hold. Since µ∗ = ν ∗ , µ(x) = 0 ⇔ ν(x) = 0 for all x ∈ X. Suppose µ(x) > µ(y). Let t = µ(x). Then x ∈ µt and µt = ∅. Thus there is an s ∈ [0, 1] such that µt = νs . Suppose ν(y) ≥ ν(x). Then ν(y) ≥ ν(x) ≥ s since x ∈ µt = νs . Thus y ∈ νs = µt , a contradiction. Hence ν(x) > ν(y). Similarly, ν(x) > ν(y) implies µ(x) > µ(y). Hence µ ∼ ν. In order to prove µ ≈ ν, it remains to be shown that µR = ν R . Since µ ∼ ν, it follows for all x, y ∈ X that µ(x) ≥ µ(y) if and only if ν(x) ≥ ν(y) and that µ(x) = 0 if and only if ν(x) = 0. Thus µµ(y) = νν(y) and {y ∈ X | µ(y) = µ(x)} = {y ∈ X | ν(y) = ν(x)} for all x ∈ X. Let z ∈ µR . Then for all t > µ(z), there exists y ∈ X such that µ(y) > µ(z) and µ(y) < t. Thus µt = µµ(z) \{y ∈ X | µ(y) = µ(z)}. Suppose t ≤ µ(z). Then µt = µµ(z) \{y ∈ X |µ(y) = µ(z)} since {y ∈ X | µ(y) = µ(z)} = ∅. Suppose ν(z) ∈ / ν R . Then there exists s ∈ [0, 1] such that s > ν(z) and for all x ∈ X, ν(x) > ν(z) ⇔ ν(x) ≥ s. Hence νs = νν(z) \{y ∈ X | ν(y) = ν(z)} = µµ(z) \{y ∈ X | µ(y) = µ(z)}. However there is no t ∈ [0, 1] such that µt = νs , a contradiction. Thus z ∈ ν R . Therefore, µR ⊆ ν R . By symmetry, ν R ⊆ µR. Consequently, µ ≈ ν. Conversely, suppose µ ≈ ν. Then µ ∼ ν and µ∗ = ν ∗ . Let a = ∨{µ(x) | x ∈ X} and b = ∨{ν(x) | x ∈ X}. If t > a, then µt = ∅. Thus we only need to consider the case t ≤ a. Since µ ∼ ν, we have that µ(x) = a ⇔ ν(x) = b. If t = a and µt = ∅, then there exists x0 ∈ X such that µ(x0 ) = a = t. Let s = b. Then clearly µt = νs since µ(x) = a ⇔ ν(x) = b. For t < a, we consider t1 = ∧{µ(x) | x ∈ µt }. The case when µ(z) = t1 for some z ∈ X has been considered in Proposition 8.1.5. Thus suppose t1 = µ(x) for all x ∈ X. Then µt = µt1 = µ−1 (t1 , 1] = µ−1 (t, 1]. Let s = ∧{ν(x) | x ∈ µt }. Then it is shown in Proposition 8.1.5 that µt = ν −1 (s, 1] ⊆ ν s . If for any x ∈ X, ν(x) = s, then ν −1 (s, 1] = νs and so µt = νs . If there exists z ∈ X such that ν(z) = s and z ∈ µt , then {x ∈ X | ν(x) = s} ⊆ µt since µ ∼ ν. Thus µt = νs . Suppose / ν R , then s ∈ {ν(x) | x ∈ µt } since s = ∧{ν(x) | x ∈ µt }. z∈ / µt . If s = ν(z) ∈ Thus there exists x0 ∈ µt such that ν(x0 ) = s = ν(z) and so µ(z) = µ(x0 ) ≥ t since µ ∼ ν. Hence z ∈ µt , a contradiction. Thus s = ν(z) ∈ ν R and so µ(z) ∈ µR . For all x ∈ µt , ν(x) > s and so µ(x) > µ(z) since t1 = µ(z). Hence µ(z) < t1 . Since µ(z) ∈ µR , there exists y ∈ X such that µ(z) < µ(y) < t1 and y∈ / µt = ν −1 (s, 1]. However, ν(y) > ν(z) = s and so y ∈ ν −1 (s, 1]. However, this is impossible. Thus z ∈ / µt doesn’t hold. Therefore, µt = νs .

8.2 Fuzzy Subgroups of p-groups Let G denote a ﬁnite Abelian group. If µ is a fuzzy subgroup of G, we assume that µ(0) = 1, where 0 is the identity of G. Then the only allowable fuzzy subgroup µ of a trivial group is such that µ(0) = 1. For a prime p, a ﬁnite group G is called a p-group if the order of G is a power of p. A cyclic p-group is isomorphic to Zpn for some positive integer n. Let k be a positive integer. For the cyclic group Zpn , we write Zpn ⊃ Zpk , k < n, to mean that Zpn contains the cyclic subgroup pn−k of order pk . For example, by Z33 ⊃ Z32 we mean

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Z33 ⊃ 3. We say two fuzzy subgroups are distinct if they are not equivalent. A maximal chain Zpn ⊃ Zpn−1 ⊃ ... ⊃ Zp ⊃ 0 deﬁnes a fuzzy subgroup µ as follows: µ assumes tn on Zpn \Zpn−1 , tn−1 on Zpn−1 \Zpn−2 , ..., t1 on Zp \{0} and 1 on {0}, where 1 ≥ t1 ≥ t2 ≥ ... ≥ tn−1 ≥ tn ≥ 0. The fuzzy subgroup µ just deﬁned is simply denoted by 1t1 t2 ...tn−1 tn . We refer to the heights 1, t1 , ..., tn as symbols. If n = 1, then the group is Zp . Any fuzzy subgroup of Zp is equivalent to one of the following three: 11, 1t, 10, where 1 > t ≥ 0 and the 11 represents the crisp trivial subgroup Zp and 10 represents the trivial subgroup {0} of Zp . 1t represents the fuzzy subgroup µ(x) = 1 when x = 0 and µ(x) = t when x = 0. Clearly, these are the only distinct equivalence classes of fuzzy subgroups of Zp since the only crisp subgroups of Zp are {0} and Zp itself. Example 8.2.1. Let n = 2. Then there are seven distinct equivalence classes of fuzzy subgroups of Zp2 corresponding to the maximal chain, Zp2 ⊃ Zp ⊃ {0}. These are given by the symbols 111, 11t, 110, 1tt, 1ts, 1t0, 100, where by 1ts we mean the fuzzy subgroup µ deﬁned by µ(x) = 1 when x = 0, µ(x) = t when x ∈ Zp \{0} and µ(x) = s otherwise, with 1 > t > s ≥ 0. It is easily seen that the number of fuzzy subgroups whose support is Zp2 is one more than the number of fuzzy 2 subgroups whose supports are properly contained in Zp2 . Clearly, 7 = k=0 2k = 22+1 − 1. Now 22 is the number of fuzzy subgroups whose support is Zp2 , 21 is the number of fuzzy subgroups whose support is Zp , and 20 is the number of fuzzy subgroups whose support is {0}. Example 8.2.2. Let n = 3. We consider the number of distinct equivalence classes of fuzzy subgroups of Zp3 . Consider the list of symbols in the previous example. We attach symbols 1, t, 0 to 111; t, s, 0 to 11t; 0 to 110; t, s, 0 to 1tt; s, 0, r to 1ts; 0 to 1t0; 0 to 100. This yields 1111, 111t, 1110, 11tt, 11ts, 11t0, 1100, 1ttt, 1tts, 1tt0, 1tss, 1ts0, 1tsr, 1t00, 1000. 3 Thus there are 15 distinct fuzzy subgroups of Zp3 . Clearly, 15 = k=0 2k = 23+1 − 1. Each term in the sum in LHS represents the number of fuzzy subgroups of Zp3 with speciﬁc support as described as follows: For k = 0, 1, 2, 3, it follows that 2k is the number of fuzzy subgroups whose support is Zpk .(Recall that for the cyclic group Zpn , we write Zpn ⊃ Zpk , k < n, to mean that Zpn contains the cyclic subgroup pn−k of order pk .) Proposition 8.2.3. Let n ∈ N. Then there are 2n+1 − 1 distinct equivalence classes of fuzzy subgroups of Zpn .

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Proof. We ﬁrst note that there is a one-to-one correspondence between the distinct fuzzy subgroups whose support is Zpn and distinct fuzzy subgroups whose support is properly contained in Zpn . In this correspondence, the fuzzy subgroup 111...1 is excluded. We now prove n by induction that the number of not equivalent fuzzy subroups of Zpn is k=0 2k = 2n+1 − 1. The statement is clearly true for n = 1, 2, and 3 as shown in the above examples. Assume the statement is true for n = k, i.e., the number of fuzzy subgroups of Zpk k+1 k+1 k+1 is 2k+1 − 1 = 2 2 + 2 2 − 1, the induction hypothesis. There are 2 2 not equivalent fuzzy subgroups of Zpk whose support is Zpk . Each gives rise to two fuzzy subgroups of Zpk+1 whose support is Zpk+1 and one fuzzy subgroup k+1 whose support is Zpk . Thus the 2 2 fuzzy subgroups of Zpk give rise to k+1 k+1 k+1 2( 2 2 ) + ( 2 2 ) fuzzy subgroups of Zpk+1 . The remaining 2 2 − 1 fuzzy subgroups of Zpk have supports that are properly contained in Zpk and thus k+1 yield 2 2 −1 fuzzy subgroups of Zpk+1 by simply attaching zero to each. Thus k+1 k+1 the number of not equivalent fuzzy subgroups of Zpk+1 is 2( 2 2 ) + 2 2 + 2k+1 k+1 + 2k+1 − 1 = 2 · 2k+1 − 1 = 2(k+1)+1 − 1. 2 −1=2 We now consider Zp ⊕ Zq , where p and q are distinct primes. We characterize all equivalent fuzzy subgroups of Zpn ⊕ Zq . Let n = 1. We see that Zp ⊕ Zq has the following maximal chains each of which can be identiﬁed with the chain Zp2 ⊃ Zp ⊃ {0}, namely, Zp ⊕ Zq ⊃ Zp ⊕ {0} ⊃ {0} and Zp ⊕ Zq ⊃ {0} ⊕ Zq ⊃ {0}. Each of these chains yield 7 distinct fuzzy subgroups according to Example 8.2.1. Of these 7, three yield identical fuzzy subgroups viz. 111, 1tt, 100. The other 4 viz. 11t, 110, 1ts, 1t0 yield distinct fuzzy subgroups as can easily be checked by writing it out fully. Therefore Zp ⊕ Zq has 11 nonequivalent fuzzy subgroups. Hence each one of Z6, Z15 , Z35 has 11 distinct fuzzy subgroups. Similarly, we can determine distinct fuzzy subgroups of Zp2 ⊕ Zq by considering the three maximal chains each of which is equivalent to the chain Zp3 ⊃ Zp2 ⊃ Zp ⊃ {0}. Each chain yields 23+1 − 1 = 15 fuzzy subgroups according to Example 8.2.2. Of these 15, three yield identical fuzzy subgroups as above and 8 yield two distinct fuzzy subgroups and 4 yield three distinct fuzzy subgroups. Thus we have 3 · 1 + 8 · 2 + 4 · 3 = 31 distinct fuzzy subgroups of Zp2 ⊕ Zq . For example, Z12 , Z18 , Z20 each have 31 distinct fuzzy subgroups. More generally, we have the following result. Theorem 8.2.4. Zpn ⊕ Zq has 2n+1 (n + 2) − 1 distinct fuzzy subgroups. Proof. We prove the result by induction on n. We have seen that the statement is true for the ﬁrst few n. Assume now that the statement is true for n = k, that is, Zpk ⊕Zq has 2k+1 (k +2)−1 distinct fuzzy subgroups. We want to show that Zpk+1 ⊕ Zq has 2k+2 (k + 3) − 1 distinct fuzzy subgroups. The number of distinct fuzzy subgroups with Zpk ⊕ Zq as support is 2k (k + 2). Each of these fuzzy subgroups yield three further distinct fuzzy subgroups and the rest of

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the fuzzy subgroups have support strictly contained in Zpk ⊕Zq and thus yield 2k (k + 2) − 1 distinct fuzzy subgroups. Now Zpk+1 ⊕ Zq has one maximal chain more than Zpk ⊕ Zq . This chain is Zpk+1 ⊕ Zq ⊃ Zpk+1 ⊕ {0} ⊃ · · · ⊃ {0}. The only fuzzy subgroups from this chain that are not listed from the previous chains are those that have distinct symbols on Zpk+1 ⊕{0} and Zpk+1 ⊕Zq . This set of fuzzy subgroups will come from the set of fuzzy subgroups corresponding to the maximal chain of length k +2. Now the number of such fuzzy subgroups is [(2k+3 − 1) + 1]/2 = 2k+2 since the total number of distinct fuzzy subgroups of a maximal chain of length k +2 is 2k+3 −1. Combining, we have the number of distinct fuzzy subgoups in total to be 3 × 2k (k + 2) + 2k (k + 2) − 1 + 2k+2 = 2k+2 (k + 3) − 1. Thus the induction is complete. The next case that we are interested in, is the number of possible distinct fuzzy subgroups of Zp ⊕ Zp , where p is prime. However, ﬁrst we need to determine the number of maximal chains of Zp ⊕ Zp . Theorem 8.2.5. Let p be a prime. Then the number of maximal chains of G = Zp ⊕ Zp is p + 1. Proof. Since the order of G is p2 , every nontrivial subgroup of G is cyclic and of order p. Thus every maximal chain of subgroups in G is of the form G ⊃ < a > ⊃ {0}, where a is of the form (i, j) for 1 ≤ i, j < p. The distinct subgroups generated by a are all given by (1, p) , (p, 1) , (1, 1) , (1, 2) , (1, 3) , ..., (1, p − 2) , (i, p − i) for any ﬁxed i strictly between 1 and p − 1. The subgroup generated by (i, j) with i = j and i + j = p contains one of (1, 2) , (1, 3) , ..., (1, p − 2) . The case i = j is covered by (1, 1) and the case i + j = p is covered by (i, p − i) . The only two other cases that are not covered by these cases are (1, p) and (p, 1) which are listed above. Proposition 8.2.6. For p a prime, the number of distinct fuzzy subgroups of G = Zp ⊕ Zp is 4p + 7. Proof. Since there are three levels in every maximal chain of G, we have a similar situation as in Example 8.2.1. Of the 7 possible symbols, there are 4 that will give rise to distinct fuzzy subgroups and these are of the form 11t, 110, 1ts, and 1t0. Thus by Theorem 8.2.5, there are 4 (p + 1) distinct fuzzy subgroups for this situation. The rest of the symbols each give rise to one fuzzy subgroup. Hence there are 4 (p + 1) + 3 = 4p + 7 distinct fuzzy subgroups. The notion of equivalence can be viewed as a special case of the following notion of fuzzy isomorphism: Let G be a group and µ and ν be fuzzy subgroups of G. If there exists an isomorphism f : µ∗ → ν ∗ such that µ(a) > µ(b) ⇔ ν(f (a)) > ν(f (b)) where a, b ∈ µ∗ , then µ is said to be fuzzy isomorphic

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to ν and we write µ ν. It is clear that the notion of fuzzy isomorphism is an equivalence relation on the family of all fuzzy subgroups of G. The notion of equivalence of fuzzy subgroups as we have deﬁned is ﬁner than the usual notion of fuzzy isomorphism of two fuzzy subgroups as deﬁned above. That is, if two fuzzy subgroups are equivalent then they are fuzzy isomorphic, but not vice versa as illustrated by the following example. Example 8.2.7. Let G = Z2 ⊕ Z2 . Deﬁne fuzzy subgroups µ and ν of G as follows: ∀x ∈ G,  1 if x = (0, 0), µ(x) = t if x ∈ Z2 ⊕ {0}\{(0, 0)},   s otherwise;  1 if x = (0, 0), ν(x) = t if x ∈ {0} ⊕ Z2 \{(0, 0)},  s otherwise, where t > s. The function on G given by (a, b) → (b, a) is a fuzzy isomorphism. Thus µ ν, but µ is not equivalent to ν. We also note that if µ and ν are fuzzy subgroups of G such that ν is deﬁned as in Example 8.2.7, µ(0, 0) = µ(1, 0) = 1 and µ(x) = t otherwise, then µ is not isomorphic to ν although their supports are isomorphic. To see this, suppose µ and ν are isomorphic and let f : µ∗ → ν ∗ be an isomorphism. Then µ(1, 0) = 1 = µ(0, 0) implies ν(f (1, 0)) = 1 = ν(0, 0). Thus f (1, 0) = (1, 1) or (1, 0). Thus we have f (1, 0) = (0, 1) which implies ν(0, 1) = 1, a contradiction. If µ ν, then |Im(µ)\{1}| = |Im(ν)\{1}|. We have shown previously that Z22 has 7 distinct (nonequivalent) fuzzy subgroups, namely, 111, 11t, 110, 1tt, 1ts, 1t0, 100. The three maximal chains of Z2 ⊕ Z2 yield 15 distinct fuzzy subgroups of Z2 ⊕Z2 . However, Z2 ⊕Z2 has 7 non-isomorphic fuzzy subgroups. For example, the µ and ν deﬁned in the previous example are fuzzy isomorphic, implying that the fuzzy subgroup 1ts counts only once in the case of isomorphism. However, it counts three times in the case of equivalence.

8.3 Pad Keychains We now consider the problem of determining the number of distinct equivalence classes of fuzzy subgroups of G = Zp1 ⊕ ... ⊕ Zpn , where p1 , p2 , ..., pn are distinct primes. We introduce the notion of a keychain of a chain of length n + 1 and the index of a keychain in order to determine the number of fuzzy subgroups of G. We use the terminology as introduced in [10]. In the previous section, we examined Zpn , Zp ⊕ Zp , and Zpn ⊕ Zq where p and q are distinct primes. We now investigate the number of fuzzy subgroups of G of the form Zp1 ⊕ ... ⊕ Zpn for distinct primes p1 , p2 , ..., pn . Since the

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membership values of fuzzy subgroups of ﬁnite groups form a ﬁnite chain, we consider chains of real numbers in [0, 1] each of length n + 1. For the equivalence relation ∼, we denote the equivalence class containing µ by [µ]. A ﬁnite n-chain is a collection of numbers in [0, 1] of the form 1 > t1 > t2 > ... > tn−1 > tn , where tn may or may not be zero. This is written simply as 1t1 t2 ...tn in descending order. The numbers 1, t1 , ..., tn−1 , tn are called pins . We say that 1 occupies the ﬁrst position and ti occupies the (i + 1)-th position for i = 1, ..., n. The length of an n-chain is n + 1. Thus the number of positions available in an n-chain is equal to the length of the chain which is n + 1. The positions play a crucial role in the investigation. Unless otherwise stated, we ﬁx n. An n-chain is called a keychain if 1 ≥ t1 ≥ t2 ≥ ... ≥ tn−1 ≥ tn . A consecutive occurrence of equality signs is said to be an interlocking position of pins. Interlocked pins are called components. However, a 1 standing alone in the ﬁrst position not interlocked with any other t’s is not considered to be a component of any keychain. A k-pad (1 ≤ k ≤ n) is a keychain that contains k distinct components. For example, 1 > t1 = t2 > t3 = t4 = t5 > t6 > 0 is a 4-pad keychain of a 7-chain, whereas 1 = t1 = t2 = t3 > t4 > t5 = t6 = t7 is a 3-pad keychain of a 7-chain. The number of pins found in the interlocked position forming a component is called the padidity of the component. Thus in our example of a 4-pad keychain, the padidities of the components are respectively 2, 3 and 1. The index of a k-pad keychain is deﬁned to be the set of padidities of various components of the keychain in which singleton components are ignored for the sake of simplicity. Thus the index of a keychain forms a ‘partition’ of the number n. In our example, for n = 7 and k = 4, the partition of 7 given by 2 + 3 + 1 + 1 corresponds to a 4-pad keychain whose index is (2, 3). Another partition of 7 given by 3 + 1 + 3 corresponds to a 3-pad keychain whose index is (3, 3). More generally, by an index of a k-pad keychain we mean a ﬁnite set (unordered) of positive integers (l1, l2, ..., lk ), where li ≥ 2 for i = 1, ..., k and l1 + l2 + ... + lk ≤ n. From these examples, it we see that the index and the set of padidities determine each other. We next determine the number of keychains of a chain of length n. When we write ∗-pad, the ∗ denotes a natural number between 1 and n − 1. For any n ≥ 2, there are three 1-pad keychains. They are of the form 111...1, 1tt...t, 100...0, with the padidity being n − 1. The following propositions illustrate the inductive steps needed for proving the main result. Accordingly, in the following, we assume 2 ≤ k ≤ n − 2. Proposition 8.3.1. The number of (n − k)-pad keychains of length n with index k is 4(n − k). Proof. Since the index of any (n − k)-pad keychain is given to be k, it follows that one pin is repeated k times and the other (n − k − 1)-pins are distinct. Consider the (n − k)-pad keychain given as follows: 1 ≥ t1 = t2 = ... = tk > tk+1 > ... > tn−1 .

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8 Equivalence of Fuzzy Subgroups of Finite Abelian Groups

Corresponding to this chain, there are 4 distinct (n − k)-pad keychains: 1 = t1 = ... = tk > tk+1 > ... > tn−1 and 1 > t1 = t2 = ... = tk > tk+1 > ... > tn−1 with tn−1 > 0 and two more keychains obtained by letting tn−1 = 0. That is, there are 4(n − k) -pad keychains corresponding to t1 . Similarly, we obtain 4 distinct (n − k)-pad keychains corresponding to t2 , ..., tn−k on up to tn−k , ..., tn−1 . Clearly, these are the only possible (n − k)-pad keychains of index k, yielding a total of 4(n − k). From the above argument, it follows that there are four (n − 1)-pad keychains for any n ≥ 3. These are 1t1 t2 ...tn−1 > 0, 11t2 ...tn−1 > 0, 1t1 t2 ...tn−2 0, 11t2 ...tn−2 0. The next proposition considers the number of keychains when the index consists of two natural numbers of the form (k1 , k2 ). Proposition 8.3.2. The number of (n − k1 − k2 + 1)-pad keychains of length n with index (k1 , k2 ) is 4(n−k1 −k2 +1)! (n−k1 −k2 −1)! 4(n−k1 −k2 +1)! 2!(n−k1 −k2 −1)!

if k1 = k2 , if k1 = k2 ,

ki > 1 for i = 1, 2. Proof. Assume k1 = k2. Since the index of any (n − k1 − k2 + 1)-pad keychain is given to be (k1, k2 ), we have that one pin is repeated k1 times and the other one is repeated k2 times, while the remaining n − k1 − k2 − 1 pins are all distinct. Consider the (n − k1 − k2 + 1)-pad keychain given as follows: 1 ≥ t1 = t2 = ... = tk1 > tk1 +1 = tk1 +2 = ... = tk1 +k2 > tk1 +k2 +1 > ... > tn−1··· . We collapse the ﬁrst k1 + 1 positions into one so that t1 is the leading pin as follows: t1 > tk1 +1 = tk1 +2 = ... = tk1 +k2 > tk1 +k2 +1 > ... > tn−1··· . The length of this chain is n − k1 . By Proposition 8.3.1, it follows that the number of (n − k1 − k2 )-pad keychains of length n − k1 with index k2 is 4 (n − k1 − k2 ) . By considering the collapsed positions, it follows that the number of (n − k1 − k2 + 1)-pad keychains of length n with index (k1 , k2 ) corresponding to t1 being repeated k1 times is 4(n − k1 − k2 ). Another 4(n − k1 − k2 ) keychains are obtained by interchanging the roles of k1 and k2 in the above discussion. Thus the number of (n − k1 − k2 + 1)-pad keychains of length n with index (k1 , k2 ) corresponding to the two repetitions of t1 is 8(n − k1 − k2 ).We repeat the above process with t1 replaced by t2 . Thus if t2 is repeated k1 times, then we may have the following chain

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1 ≥ t1 > t2 = t3 = ... = tk1 +1 > tk1 +2 ≥ ... ≥ t.... If, in addition, one other pin is repeated k2 times, then one such keychain is 1 ≥ t1 > t2 = ... = tk1 +1 > tk1 +2 = ... = tk1 +k2 +1 > tk1 +k2 +2 > ... > tn−1 .... Collapsing the ﬁrst k1 + 2 positions into one, we have the chain of length n − 1 − k1 : t2 > tk1 +2 = ... = tk1 +k2 +1 > tk1 +k2 +2 > ... > tn−1 .... By Proposition 8.3.1, the number of (n − k1 − 1 − k2 )-pad keychains of length n − k1 − 1 with index k2 is 4(n − k1 − k2 − 1). By considering the collapsed positions, it follows that the number of (n−k1 −k2 +1)-pad keychains of length n with index (k1, k2 ) corresponding to the two repetitions of t2 is 8(n−k1 −1− k2 ). Proceeding inductively, we obtain 8 · 1 of (n − k1 − k2 + 1)-pad keychains of index (k1, k2 ) corresponding to the k1 and k2 repetitions of tn−k1 −k2 . (The inductive step must stop here because the subscript of t cannot be strictly less than n − k1 − k2 for any keychain of index (k1 , k2 ). Thus the total number of (n − k1 − k2 + 1)-pad keychains of index (k1 , k2 ) is 8[(n − k1 − k2 ) + (n − k1 − k2 − 1) + ... + 2 + 1]. n Using the formula, i=1 i = (n + 1)n/2, it follows that this number is equal to 4(n − k1 − k2 + 1)! . (n − k1 − k2 − 1)! We now consider the case, where k1 = k2 . Since k1 = k2 , interchanging the roles of k1 and k2 yields identical (n − k1 − k2 + 1)-pad keychains, unlike the case, where k1 = k2 . Thus the number of (n − k1 − k2 + 1)-pad keychains in this case is half the number in the case for distinct k1 , k2 . Hence we have 1 2 (8[(n

− k1 − k2 ) + (n − k1 − k2 − 1) + ... + 2 + 1])

(n − k1 − k2 + 1)-pad keychains. This latter number is equal to 1 4(n − k1 − k2 + 1)! , 2 (n − k1 − k2 − 1)! the desired number.

Proposition 8.3.3. The number of (n − k1 − k2 − k3 + 2)-pad keychains of 4(n−k1 −k2 −k3 +2)! , where m is the number length n with index (k1 , k2 , k3 ) is m!(n−k 1 −k2 −k3 −1)! of identical ki ’s, ki > 1 for i = 1, 2, 3. If k1 , k2 , k3 are all distinct, then the number of(n−k1 −k2 −k3 +2)-pad keychains of length n with index (k1 , k2 , k3 ) is

212

8 Equivalence of Fuzzy Subgroups of Finite Abelian Groups 4(n−k1 −k2 −k3 +2)! (n−k1 −k2 −k3 −1)! .

Proof. We ﬁrst consider the case where all the ki ’s are distinct. Since the index of any (n − k1 − k2 − k3 + 2)-pad keychain is given to be (k1 , k2 , k3 ), we know that one pin is repeated k1 times, another one is repeated k2 times, and a third one is repeated k3 times while the remaining n − k1 − k2 − k3 − 1 pins are all distinct. Consider the (n − k1 − k2 − k3 + 2)-pad keychain given as follows: 1 ≥ t1 = t2 = ... = tk1 > tk1 +1 = tk1 +2 = ... = tk1 +k2 > tk1 +k2 +1 = ... = tk1 +k2 +k3 > tk1 +k2 +k3 +1 > ... > tn−1 .... We collapse the ﬁrst k1 + 1 positions into one so that t1 is the leading pin as follows: t1 > tk1 +1 = ... = tk1 +k2 > tk1 +k2 +1 = tk1 +k2 +2 = ... = tk1 +k2 +k3 > tk1 +k2 +k3 +1 > ... > tn−1 .... The length of this chain is n − k1. By Proposition 8.3.2, it follows that the number of (n − k1 − k2 − k3 + 1)pad keychains of length n − k1 with index (k2 , k3 ) is 4(n − k1 − k2 − k3 + 1)(n − k1 − k2 − k3 ). By considering the collapsed positions, it follows that the number of (n − k1 − k2 − k3 + 2)-pad keychains of length n with index (k1 , k2 , k3 ) corresponding to t1 being repeated k1 times is (4n − k1 − k2 − k3 + 1)(n − k1 − k2 − k3 ). An additional 8(n − k1 − k2 − k3 + 1)(n − k1 − k2 − k3 ) keychains are obtained by interchanging the roles of k1 , k2 and k3 in the above discussion. Thus the number of (n − k1 − k2 − k3 + 2)-pad keychains of length n with index (k1 , k2 , k3 ) corresponding to the three repetitions of t1 is 12(n − k1 − k2 − k3 + 1)(n − k1 − k2 − k3 ). We repeat the above process with t1 replaced by t2 . Thus if t2 is repeated k1 times, then we may have the following chain 1 ≥ t1 > t2 = t3 = ... = tk1 +1 > tk1 +2 > ... > tn−1 .... If, in addition, one other pin is repeated k2 times, and a third one is repeated k3 times, then one such keychain is as follows: 1 ≥ t1 > t2 = ... = tk1 +1 > tk1 +2 = ... = tk1 +k2 +1 > tk1 +k2 +2 = ... = tk1 +k2 +k3 +1 > tk1 +k2 +k3 +2 > ... > tn−1 .... Collapsing the ﬁrst k1 + 2 positions into one, we have the chain of length n − 1 − k1 : t2 > tk1 +2 = ... = tk1 +k2 +1 > tk1 +k2 +2 = ... = tk1 +k2 +k3 +1 > tk1 +k2 +k3 +2 > ... > tn−1 ....

8.3 Pad Keychains

213

By Proposition 8.3.2, the number of (n−k1 −1−k2 −k3 +1)-pad keychains of length n−k1 −1 with index (k2 , k3 ) is 4(n−k1 −k2 −k3 )(n−k1 −k2 −k3 −1). By considering the collapsed positions, it follows that the number of (n − k1 − k2 − k3 + 2)-pad keychains of length n with index (k1 , k2 , k3 ) corresponding to the three repetitions of t2 is 12(n − k1 − k2 − k3 )(n − k1 − k2 − k3 − 1). Proceeding inductively, we get 12(3·2) of the (n − k1 − k2 − k3 + 2)pad keychains of index (k1 , k2 , k3 ) corresponding to the three repetitions of tn−k2 −k2 −k3 −1 being k1 , k2 , k3 , respectively. We also get 12(2·1) of the (n − k1 − k2 − k3 + 2)-pad keychains of index (k1 , k2 , k3 ) corresponding to the three repetitions of tn−k2 −k2 −k3 being k1 , k2 , k3 , respectively. (The inductive step must stop here since the subscript of t cannot be strictly less than n − k1 − k2 − k3 for any keychain of index (k1 , k2 , k3 ).) Thus the total number of (n − k1 − k2 − k3 + 2)-pad keychains of index (k1 , k2 , k3 ) is 12[(n − k1 − k2 − k3 + 1)(n − k1 − k2 − k3 ) +(n − k1 − k2 − k3 )(n − k1 − k2 − k3 − 1 + ... + 3 · 2 + 2 · 1)]. By using the formula n(n+1)+(n−1)n+...+2.3+1.2 = 13 n(n+1)(n+2), it follows that this number is equal to 4(n−k1 −k2 −k3 +2)! (n−k1 −k2 −k3 −1)!

.

We now consider the case where exactly two of k1 , k2 , k3 are equal. It follows as in Proposition 8.3.2 that dividing this number by 2! yields the desired number. If all three of the ki ’s are equal, then the desired number is obtained by dividing by 3!. We use the following notation for the general case. Suppose S = (k1 , k2 , ..., km ) is the given index of a keychain, where not all ki ’s need to be distinct. Then S can be partitioned by a set {F1 , F2 ..., Fs } of equivalence classes induced by an equivalence relation ∼ on {k1 , k2 , ..., km } given by ki ∼ kj if and only if ki = kj , 1 ≤ i, j ≤ m. Let li denote the number of elements in the class Fi for each i = 1, 2, ..., s. The next theorem deals with the most general case. Theorem 8.3.4. The number of (n − k1 − k2 − ... − km + m − 1)-pad keychains of length n with index (k1 , k2 , ..., km ) is 4(n−k1 −k2 −...−km +m−1)! (n−k1 −k2 −...−km −1)!

if all the padidities ki are distinct, otherwise the number is 4(n−k1 −k2 −...−km +m−1)! l1 !l2 !...ls !(n−k1 −k2 −...−km −1)! .

Proof. Suppose all the ki ’s are distinct. Then the result is true for m = 1, 2 and 3, respectively. Assume the result is true when the number of ki ’s is m−1, the induction hypothesis. Consider the (n − 1)-chain

214

8 Equivalence of Fuzzy Subgroups of Finite Abelian Groups

1 ≥ t1 ≥ t2 ≥ ... ≥ tn−1 . Suppose t1 is repeated k1 times. Then, we have the chain 1 ≥ t1 = t2 = ... = tk1 > tk1 +1 ≥ ... ≥ tn−1 . Now collapse the ﬁrst k1 + 1 positions so that t1 is a leading pin as follows: t1 > tk1 +1 ≥ ... ≥ tn−1 . This is a keychain of length (n − k1 ). From this chain, it follows that the number of (n − k1 − k2 − ... − km + m − 2)-pad keychains of length n − k1 with index (k2 , k3 , ..., km ) is 4(n−k1 −k2 −...−km +m−2)! (n−k1 −k2 −...−km −1)! .

By considering the collapsed positions, it follows that the number of (n − k1 − k2 − ... − km + m − 1)-pad keychains of length n with index (k1 , k2 , .., km ) corresponding to t1 being repeated k1 times is 4(n−k1 −k2 −...−km +m−2)! (n−k1 −k2 −...−km −1)! . 1 −k2 −...−km +m−2)! We obtain an additional 4(m − 1) (n−k keychains by in(n−k1 −k2 −...−km −1)! terchanging the roles of k1 , k2 , ..., and km in the above discussion. Thus the number of (n − k1 − k2 − ... − km + m − 1)-pad keychains of length n with index (k1 , k2 , ..., km ) corresponding to the m repetitions of t1 is 1 −k2 −...−km +m−2)! 4m (n−k (n−k1 −k2 −...−km −1)! .

We repeat the above process with t1 replaced by t2. Thus if t2 is repeated k1 times, then the ﬁrst chain becomes 1 ≥ t1 > t2 = t3 = ... = tk1 +1 > tk1 +2 ≥ ... ≥ tn−1 . The keychain t2 > tk1 +2 ≥ ... ≥ tn−1 is of length (n − k1 − 1). From this chain, the induction hypothesis assures us that the number of (n − k1 − 1 − k2 − ... − km + m − 2)-pad keychains of index (k2 , k3 , · · · , km ) is 1 −1−k2 −...−km +m−2)! 4 (n−k . (n−k1 −k2 −...−km −2)!

Combining this with the chain 1 ≥ t1 > t2 = t3 = ... = tk1 +1 > tk1 +2 ≥ ... ≥ tn−1 and using the previous discussion, it follows that the number of (n − k1 − k2 − ... − km + m − 1)-pad keychains of length n with index (k1 , k2 , ..., km ) corresponding to the m repetitions of t2 is 1 −k2 −...−km +m−3)! 4m (n−k (n−k1 −k2 −...−km −2)! .

8.3 Pad Keychains

215

Proceeding inductively, corresponding to tn−k1 −k2 −...−km −1 being repeated k1 , k2 , ..., km times respectively, the number of (n − k1 − k2 − ... − km + m − 1)pad keychains of index (k1 , k2 , ..., km ) is 4m(m!). Also, there are 4m(m − 1)! similar keychains corresponding to the repetitions of tn−k1 −k2 −...−km . (The inductive step must stop here since the subscript of t cannot be strictly less than n−k1 −k2 −...−km for any keychain of index (k1 , k2 , ..., km ).) Thus the total number of (n − k1 − k2 − ... − km + m − 1)-pad keychains of index (k1 , k2 , ..., km ) is 4m[(n − k1 − k2 − ... − km + m − 2)(n − k1 − k2 − ... − km + m − 3) · · · (n − k1 − k2 − ... − km )+ (n − k1 − k2 − ... − km + m − 3)...(n − k1 − k2 − ... − km − 1) + ... + m(m − 1)...2 + (m − 1)(m − 2)...2 · 1] which can be easily seen to be equal to 4(n−k1 −k2 −...−km +m−1)! (n−k1 −k2 −...−km −1)! , where we have used the fact that n(n + 1)...(n + k − 1 n(n + 1)(n + 2)...(n + k). 1) + ... + 2 · 3...(k + 1) + 1 · 2...k = k+1 Now consider the case, where some ki ’s are equal. Then as in Proposition 8.3.3, the desired number is obtained by dividing the expression above by the factorials of the numbers of identical ki ’s. Corollary 8.3.5. If n = k1 + k2 + · · · + km + s + 1, then the number of (n−k1 −k2 −...−km +m−1)-pad keychains of length n with index (k1 , k2 , ..., km ) is 4(s+m)! s!

if all the ki are distinct, where s is the number of nonrepeating pins from the second position on. Otherwise, the number is 4(s+m)! s!l1 !l2 !···lt ! ,

where l1 , l2 , . . . , lt are the numbers of identical ki ’s, ki > 1, for i = 1, 2, ..., m. Proof. Suppose all the ki ’s are distinct. By Theorem 8.3.4, the desired number is 1 −k2 −...−km +m−1)! 4 (n−k (n−k1 −k2 −...−km −1)! .

Clearly, s + 1 = n − k1 − k2 − ... − km . Thus the desired number is 4 (s+1+m−1)! = 4 (s+m)! . (s+1−1)! s! The case when some ki ’s are identical follows by arguments similar to those previously given. Example 8.3.6. (1) We detertmine the number of 4 -pad keychains of length 8 with index (2, 2, 2). By Corollary 8.3.5, we have 8 = 2 + 2 + 2 + 1 + 1 so that s = 1, m = 3, l1 = 3. Hence the number of (8 − 2 − 2 − 2 + 3 − 1)-pad = 16. keychains of the given index is 4 (3+1)! 3!

216

8 Equivalence of Fuzzy Subgroups of Finite Abelian Groups

The chain 1 ≥ t1 = t2 > t3 = t4 > t5 = t6 > t7 is a representative 4-pad keychain of length 8 with index (2, 2, 2). Note that s = 1 since t7 is the only nonrepeating pin from position 2 on. (2) We now determine the number of 4-pad keychains of length 10 with index (2, 2, 2, 3). By Corollary 8.3.5, we have 10 = 2 + 2 + 2 + 3 + 0 + 1 and 4 = 10 − 2 − 2 − 2 − 3 + 4 − 1. Hence m = 4, l1 = 3 and s = 0. Thus the desired number is 4 (4!) 3! = 16. Note that all pins (from the second position) repeat, hence s = 0. (3) We now ﬁnd the number of 5-pad keychains of length 11 with index (2, 2, 3). As before, we have 11 = 2 + 2 + 3 + 1 + 1 + 1 + 1. Hence m = 3, l1 = 2 and s = 3. Thus the desired number is 4 (3+3)! 3!2! = 240. Recall that any fuzzy subgroup of G = Zp is equivalent to one of the following three: 11, 1t, 10 where the 11 represents the crisp trivial subgroup Zp and 10 represents the trivial subgroup {0} of Zp . 1t represents the fuzzy subgroup µ(x) = 1 if x = 0 and µ(x) = t if x = 0∀x ∈ G. Clearly, these are the only distinct equivalence classes of fuzzy subgroups of Zp since the only crisp subgroups of Zp are {0} and Zp itself. Here we always assume 1 > t ≥ 0. Proposition 8.3.7. Let G = Zp1 ⊕...⊕Zpn−1 , where the pi are distinct primes and n ∈ N with n ≥ 3. Then the number of distinct fuzzy subgroups of G represented by one (n − 1)-chain of length n with index k is (n−1)! k! . Proof. Let Gi = Zp1 ⊕ ... ⊕ Zpi ⊕ {0} ⊕ ... ⊕ {0}. Consider a fuzzy subgroup µ of G deﬁned by µ(x) = ti for x ∈ Gi \Gi−1 with all the ti distinct except for i = j in which case j appears k times and t0 = 1. We may represent µ as 1t1 t2 ...tj−1 tj ...tj tj+k ...tn−1 . We may also write,  1 if x = 0     t1 if x ∈ G1 \{0}     if x ∈ G2 \G1 t2     ...  µ(x) = tj−1 if x ∈ Gj−1 \Gj−2   tj if Gj+k−1 \Gj−1     t if x ∈ Gj+k \Gj+k−1  j+k    ...    otherwise. tn−1 If we replace G1 by a subgroup G1k = {0} ⊕ ... ⊕ Zk ⊕ {0} ⊕ ... ⊕ {0} isomorphic to G1 , then another fuzzy subgroup µ not equivalent to the previous one is deﬁned. Since there are n − 2 distinct subgroups isomorphic to G1, there are n − 1 distinct fuzzy subgroups of the form µ. Similarly, with G1 ﬁxed, we may replace G2 by G2k = Z1 ⊕ {0} ⊕ ... ⊕ {0} ⊕ Zk ⊕ {0} ⊕ ... ⊕ {0}. This yields an additional n − 2 nonequivalent fuzzy subgroups. Repeating the process with Gj−1 , an additional n−j +1 distinct fuzzy subgroups arise. Next

8.3 Pad Keychains

217

n−j more fuzzy subgroups. Finally, k Gn−1 yields 1 fuzzy subgroup. By the Fundamental Principle of Counting, we have (n−1)! distinct fuzzy subgroups. k! we replace Gj with Gjk . This gives

Corollary 8.3.8. Let n = k + m + 1. Let G = Zp1 ⊕ ... ⊕ Zpn−1 . Then the number of distinct fuzzy subgroups of G represented by all (n − 1)-chains each of length n with index k, is = 4 (m+1)!(n−1)! . 4 (n − k) (n−1)! k! m!k! Proof. The desired result follows immediately from Proposition 8.3.7, Proposition 8.3.1 and Corollary 8.3.5. Proposition 8.3.9. Let G = Zp1 ⊕...⊕Zpn−1 , where the pi are distinct primes and n ∈ N with n ≥ 3. Then the number of distinct fuzzy subgroups represented by one (n − 1)-chain of length n with index (k1 , k2 ) is (n−1)! k1 !k2 !

if k1 = k2 . If k1 = k2 , then

(n−1)! 2k1 !k2 !

is the desired number.

Proof. The proof follows as in Proposition 8.3.7 except that here there are two n−j factors of the form . Using the Fundamental Principle of Counting ks as in Proposition 8.3.7, the desired result follows. Proposition 8.3.9 is easily extended to the general case when the index of the fuzzy subgroup is (k1 , k2 , ..., km ). Corollary 8.3.10. Let n = k1 +k2 +...+ km +s+1. Let G = Zp1 ⊕...⊕Zpn−1 . Then the number of distinct fuzzy subgroups represented by all (n − 1)-chains each of length n with index (k1 , k2 , ...km ) is 4 (s+m)!(n−1)! s!k1 !k2 !...km ! if all the ki are distinct, where s is the number of nonrepeating pins from the 2nd position. If some ki ’s are identical, divide the ﬁrst expression by the factorials of the numbers of the identical ki ’s. Example 8.3.11. We ﬁnd the number of distinct fuzzy subgroups of Zp1 ⊕ ... ⊕ Zp6 represented by all 6-chains of length 7 with index (1) 3, (2) 2 · 2, (3) 2 · 2 · 2, (4) 2 · 3. By Corollary 8.3.10, we have (1) n = 7 = 3 + 3 + 1 and s = 3, k1 = 3, m = 1 and 3 < 7 − 1. Thus the desired number is 4 (3+1)!(7−1)! = (4 · 4!6!)/(6 · 6) = 1920. 3!3!

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8 Equivalence of Fuzzy Subgroups of Finite Abelian Groups

(2) n = 7 = 2 + 2 + 2 + 1 and s = 2, kk2 = 2, l = 2 = number of identical ki ’s, m = 2 and 2 < 7 − 1. Hence the desired number is = 4(24)(6)(120)/16 = 4320. 4 (2+2)!(7−1)! 2!2!2!2! (3) n = 7 = 2 + 2 + 2 + 0 + 1 and s = 0, m = 3, k1 = k2 = k3 = 2, l = 3. Thus the desired number is 4 (0+3)!(7−1)! = 360. 0!2!2!2!3! (4) n = 7 = 2 + 3 + 1 + 1 and s = 1, m = 2. Hence the required number is (1+2)!(7−1)! 2!3!

= 1440.

8.4 Zpn ⊕ Zq m In this section, we determine the number of distinct equivalence classes of fuzzy subgroups of G = Zpn ⊕ Zqm where p and q are distinct primes and n, m ∈ N. We ﬁrst derive a formula for the number of maximal chains of G. We then derive a combinatorial formula for the number of distinct fuzzy subgroups of G. We utilize the complete characterization of ﬁnite Abelian groups in the crisp case [3]. Throughout this section, we denote G simply by pn q m and their subgroups H = Zpi ⊕Zqj in a similar fashion pi q j for 1 ≤ i ≤ n and 1 ≤ j ≤ m. (Recall that for the cyclic group Zpn , we write Zpn ⊃ Zpk , k < n, to mean that Zpn contains the cyclic subgroup of order pk generated by pn−k · 1.) For any n ∈ N and m = 0, it is clear that G has only one maximal chain, namely Zpn ⊃ Zpn−1 ⊃ ... ⊃ Zp ⊃ {0}. Proposition 8.4.1. Let G = Zpn ⊕ Zq where p and q are distinct primes. Then the number of maximal chains of G is n + 1. Proof. The proof is by induction on n. Let n = 1. Then G has two maximal chains, namely G = Zp ⊕ Zq ⊃ {0} ⊕ Zq ⊃ {0} and G = Zp ⊕ Zq ⊃ Zp ⊕ {0} ⊃ {0}. Assume that Zpk ⊕ Zq has k + 1 maximal chains. Now Zpk+1 ⊕ Zq has two maximal subgroups namely Zpk ⊕ Zq and Zpk+1 ⊕ {0} has only one maximal chain. Thus Zpk+1 ⊕ Zq has k + 2 maximal chains. The desired result follows by induction. By symmetry, it follows that Zp ⊕ Zqm has m + 1 maximal chains. The factorial expressions in the next proposition are written in a form for ease of use in inductive steps later. Also, ri = 1 for all i and m ≥ 2 in the next result, where m ≥ 2 since the case for m = 1 has been considered in Proposition 8.4.1.

8.4 Zpn ⊕ Zqm

219

Proposition 8.4.2. Let G = Zp2 ⊕ Zqm , where p and q are distinct primes m−1 and m ≥ 2. Then the number of maximal chains of G is i=−1 ri (m − i), (2+i−1)! . where ri = (2−2)!(1+i)! Proof. The proof is by induction on m. Let m = 2. Then G = Zp2 ⊕ Zq2 has 6 maximal chains, namely p2 q 2 ⊃ p2 q ⊃ pq ⊃ p ⊃ 0 p2 q 2 ⊃ p2 q ⊃ pq ⊃ q ⊃ 0 p2 q 2 ⊃ p2 q ⊃ p 2 ⊃ p ⊃ 0 p2 q 2 ⊃ pq 2 ⊃ pq ⊃ p ⊃ 0 p2 q 2 ⊃ pq 2 ⊃ pq ⊃ q ⊃ 0 p2 q 2 ⊃ pq 2 ⊃ q 2 ⊃ q ⊃ 0. Now

2−1

ri (2 − i) = 6.

i=−1

Assume that the result is true k for k ∈ N, k ≥ 2, the inductive hypothesis. We show that Zp2 ⊕ Zqk+1 has i=−1 ri (k + 1 − i) maximal chains. The number of maximal chains of Zp2 ⊕ Zqk+1 can be determined from Zp2 ⊕ Zqk . k−1 Now p2 q k has i=−1 ri (k − i) maximal chains, pq k has k + 1 maximal k−1 chains, and q k+1 has one maximal chain. Thus Zp2 ⊕Zqk+1 has i=−1 ri (k − i) k plus k + 1 + 1 maximal chains which is equal to i=−1 ri (k + 1 − i) . Thus the desired result follows by induction. n−1 By symmetry, it follows that Zpn ⊕ Zq2 has i=−1 ri (n − i) maximal chains. In order to help the reader understand the inductive process of ﬁnding the maximal chains of G = Zpn ⊕ Zqm in general, we present an additional case. Proposition 8.4.3. Let G = Zp3 ⊕ Zqm , where p and q are distinct primes. m−1 Then the number of maximal chains of G is i=−1 ri (m − i) , where ri = (3+i−1)! (3−2)!(1+i)! , m ≥ 2. Proof. Let m = 2. There are 10 maximal chains of p3 q 2 . This can be seen as follows: From Proposition 8.4.2, six of them are in the form

220

8 Equivalence of Fuzzy Subgroups of Finite Abelian Groups

p3 q 2 ⊃ p2 q 2 ⊃ p2 q ⊃ pq ⊃ p ⊃ 0 p3 q 2 ⊃ p2 q 2 ⊃ p2 q ⊃ pq ⊃ q ⊃ 0 p3 q 2 ⊃ p2 q 2 ⊃ p2 q ⊃ p 2 ⊃ p ⊃ 0 p3 q 2 ⊃ p2 q 2 ⊃ pq 2 ⊃ pq ⊃ p ⊃ 0 p3 q 2 ⊃ p2 q 2 ⊃ pq 2 ⊃ pq ⊃ q ⊃ 0 p3 q 2 ⊃ p2 q 2 ⊃ pq 2 ⊃ q 2 ⊃ q ⊃ 0. another three in the form p3 q 2 ⊃ p3 q ⊃ p2 q ⊃ pq ⊃ p ⊃ 0 p3 q 2 ⊃ p3 q ⊃ p2 q ⊃ pq ⊃ q ⊃ 0 p3 q 2 ⊃ p 3 q ⊃ p 2 q ⊃ p 2 ⊃ p ⊃ 0 and one more in the form p 3 q 2 ⊃ p3 q ⊃ p 3 . . .

.

It is easily veriﬁed that the number 10 is obtained upon substitution of m = 2 in the formula. The inductive step can be proved in a manner similar as in the proof of Proposition 8.4.2. By symmetry, we can interchange p and q in G without aﬀecting the number of maximal chains. The above propositions illustrate how the inductive proofs can be employed eﬀectively using extended diagrams from the k-level to the (k + 1)-level. Theorem 8.4.4. Let G = Zpn ⊕Zqm , where p and q are distinct primes. Then the number of maximal chains of G is m−1 i=−1

ri (m − i) , where ri =

(n + i − 1)! , n ≥ 2. (n − 2)! (1 + i)!

Proof. The proof is ny induction on m with n ﬁxed. For m = 2, 3, the theorem holds by Propositions 8.4.2 and 8.4.3. Assume that Zpn + Zqk has k−1 (n+i−1)! i=−1 ri (k − i) maximal chains, where ri = (n−2)!(1+i)! , the induction hyk pothesis. We show that Zpn + Zqk+1 has i=−1 ri (k + 1 − i) maximal chains (n+1−1)! . where ri = (n−2)!(1+i)! As in Propositions 8.4.2 and 8.4.3, it follows that (n, k) yields k−1 (n+i−1)! i=−1 rni (k − i) maximal chains, where rni = (n−2)!(1+i)! . The cases for ((n − 1) , k) , ..., (0, (k + 1)) are similar. Thus the total number of maximal chains is

8.4 Zpn ⊕ Zqm n−2 k−1

r(n−j)i (k − i) + k + 1 + 1

j=0 i=−1

=

=

n−2 k−1

(n − j + i − 1)! (k − i) +k+1+1 (n − j − 2)! (1 + i)! j=0 i=−1

n−2

[k + 1 + (n − j − 1) k +

j=0

(n − j) (n − j − 1) (k − 1) 2!

(n + k − j − 2) ... (n − j − 1) ] + (k + 1) + 1 k! = (n − 1) (k + 1) + k [(n − 1) + (n − 2) + ... + 1] +

+

n−2 (k − 1) (n − j) (n − j − 1) + ... 2! j=0

+

n−2 (k − 2) (n − j + 1) (n − j) (n − j − 1) + ... 3! j=0

+k + 1 + 1. For all n, k ∈ N, n (n + 1) ... (n + k − 1) + ... + 2.3.... (k + 1) + 1.2...k 1 n (n + 1) (n + 2) ... (n + k) . = k+1 Thus with n replaced by n − 1 and for k = 2, 3, ..., we have (n − 1) n (n + 1) ; 3 (n − 1) n (n + 1) (n + 2) ; (n + 1) n (n − 1) + ... + 3.2.1 = 4

n (n − 1) + (n − 1) (n − 2) + ... + 2.1 =

and so forth. Thus the total number of maximal chains is n (n − 1) (k − 1) (n − 1) n (n + 1) + 2 2! 3 (k − 2) (n − 1) n (n + 1) (n + 2) + 3! 4 (k − 3) (n − 1) n (n + 1) (n + 2) (n + 3) + ... + k + 1 + 1 + 4! 5 k = (n − 1) (k + 1) + n (n − 1) 2! (k − 2) (k − 1) (n − 1) n (n + 1) + (n − 1) n (n + 1) (n + 2) + 3! 4! +... + k + 1 + 1. (n − 1) (k + 1) + k

221

222

8 Equivalence of Fuzzy Subgroups of Finite Abelian Groups

Now k i=−1

ri (k + 1 − i) =

k

(n + i − 1)! (k + 1 − i) (n − 2)! (1 + i)! i=−1

= (k + 2) + (n − 1) (k + 1) +

n (n − 1) k 2!

(n + 1) n (n − 1) (k − 1) 3! (n + 2) (n + 1) n (n − 1) (k − 2) + ... + 4! (n + k − 1) (n + k − 2) ...n (n − 1) 1. + (k + 1)!

+

Clearly the expressions in the right hand side of the immediately preceding equations are equal. Thus we have the desired result. Given a maximal chain of length n + 1 of subgroups of G, there are 2n+1 distinct equivalence classes of fuzzy subgroups. However, for two such maximal chains, it is not necessarily true that the number of distinct equivalence classes of fuzzy subgroups is 2 × 2n+1 . This is because some fuzzy subgroups on distinct maximal chains determine the same equivalence class of fuzzy subgroups such as 1ttsr on the following two maximal chains p2 q 2 ⊃ p2 q ⊃ pq ⊃ p ⊃ 0, p2 q 2 ⊃ p2 q ⊃ pq ⊃ q ⊃ 0. Therefore, the number of distinct fuzzy subgroups of G = Zpn ⊕ Zqm m−1 is fewer than 2n+m+1 × i=−1 ri (m − i) , where the sum is the number of maximal chains. It is worth pointing out that the length of each maximal chain is n + m + 1. We now present a formula for the precise number of distinct fuzzy subgroups of G = Zpn ⊕ Zqm . Theorem 8.4.5. The number of distinct fuzzy subgroups of G = Zpn ⊕ Zqm is m n m 2n+m+1 2−r − 1, where m ≤ n. n − r r r=0 We prove the theorem by using a series of propositions, ﬁxing m and inducting on n. The case m = 0 is trivial, so we do not state it as a proposition but oﬀer the following brief explanation: It follows that the above formula reduces to 2n+1 20 nn m 0 − 1. In this case, there is only one maximal chain of length n + 1. By Proposition 8.2.3, the number of fuzzy subgroups is 2n+1 − 1. There is nothing to prove in this case.

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For the case m = 1, we need the following lemma. For i ∈ N, 0 ≤ i ≤ n−1, we denote the maximal chain 0 ⊂ p ⊂ p2 ⊂ ... ⊂ pi ⊂ pi q ⊂ pi+1 q ⊂ ... ⊂ pn−1 q ⊂ pn q by Ci+1 . For i = 0, we mean the maximal chain C1 given by 0 ⊂ q ⊂ pq ⊂ ... ⊂ pn−1 q ⊂ pn q. By C0 , we mean the maximal chain 0 ⊂ p ⊂ p2 ⊂ ... ⊂ pn ⊂ pn q. Lemma 8.4.6. The number of fuzzy subgroups on the maximal chain Ci+1 , 0 ≤ i ≤ n − 1, 0 ⊂ p ⊂ p2 ⊂ ... ⊂ pi ⊂ pi q ⊂ pi+1 q ⊂ ... ⊂ pn−1 q ⊂ pn q distinct from any fuzzy subgroup on any other maximal chain is 2n+1 . Proof. Consider the fuzzy subgroups on Ci+1 given by the set Si+1 of keychains 1 ≥ t1 ≥ ... ≥ ti ≥ ti+1 ≥ ti+2 ≥ ... ≥ tn+1 . It follows that there are 2n+2 distinct keychains in Si+1 . Any fuzzy subgroup µ given by an element of 2 Si+1 on Ci+1 for any ﬁxed 0 ≤ i0 ≤ n − 1 is distinct from any fuzzy subgroup on C0 since µ is represented by ... ⊂ (pi0 q)ti0 +1 ⊂ ... which cannot appear in any fuzzy subgroup on C0 . For the same reason, µ is distinct from any fuzzy subgroup on C0 . For the same reason, µ is distinct from any fuzzy subgroup on Cj for i0 + 1 < j ≤ n − 1. If 1 ≤ j ≤ i0 + 1, then µ is distinct from any fuzzy subgroup ν on Cj since ν would be represented by ... ⊂ (pj−1 q)tj ⊂ ... which cannot appear in µ. Proposition 8.4.7. The number of distinct fuzzy subgroups of G = Zpn ⊕ Zq1 is 1 n 1 2n+1+1 2−r − 1, where n ≥ 1. n−r r r=0 Proof. By Proposition 8.4.2 with m = 1, it follows that the number of maximal chains of G is equal to n + 1. Only one such maximal chain C0 goes through pn and is given by 0 ⊂ p ⊂ p2 ⊂ ... ⊂ pi ⊂ ... ⊂ pn ⊂ pn q. The length of this maximal chain is n + 2 and so the number of fuzzy subgroups on C0 is 2n+2 − 1. All other n maximal chains C1 , C2 , ..., Cn go through pn−1 q. These maximal chains can be distinguished from each other by writing pn−1 q as qpn−1 , pqpn−2 , p2 qpn−3 , ..., pn−1 q. By pi qpn−i−1 , we mean the maximal chain Ci+1 0 ⊂ p ⊂ p2 ⊂ ... ⊂ pi ⊂ pi q ⊂ pi+1 q ⊂ ... ⊂ pn−1 q. By Lemma 8.4.6, the number of fuzzy subgroups on Ci+1 is 2n+1 . Therefore, the total number of fuzzy subgroups on these n distinct maximal chains is

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8 Equivalence of Fuzzy Subgroups of Finite Abelian Groups

n2n+1 . For m = 1, the formula in Theorem 8.4.5 gives 2n+2 (1 + n2 ) − 1 which agrees with the sum of the numbers 2n+2 − 1 and n2n+1 above. Next we consider the case m = 2. First we determine the number of fuzzy subgroups on speciﬁc maximal chains. For 0 ≤ i ≤ j ≤ n − 1, let Cij denote the maximal chain, 0 ⊂ p ⊂ ... ⊂ pi ⊂ pi q ⊂ pi+1 q ⊂ ... ⊂ pj q ⊂ pj q 2 ⊂ pj+1 q 2 ⊂ ... ⊂ pn−1 q 2 ⊂ pn q 2 . For i = 0, C0j denotes the maximal chain 0 ⊂ q ⊂ pq ⊂ ... ⊂ pj q ⊂ pj q 2 ⊂ ... ⊂ pn q 2 . For j = n − 1, Ci(n−1) denotes the maximal chain Ci(n−1) 0 ⊂ q ⊂ q 2 ⊂ ... ⊂ pi q 2 ⊂ ... ⊂ pn−1 q 2 ⊂ pn q 2 . Lemma 8.4.8. The number of fuzzy subgroups on the maximal chain Cij 0 ⊂ p ⊂ ... ⊂ pi ⊂ pi q ⊂ pi+1 q ⊂ ... ⊂ pj q ⊂ pj q 2 ⊂ pj+1 q 2 ⊂ ... ⊂ pn−1 q 2 ⊂ pn q 2 distinct from any fuzzy subgroup on any other maximal chain is 2n+1 , for 0 ≤ i ≤ j ≤ n − 1. Proof. Consider the fuzzy subgroups on Cij given by the set Sij of keychains 1 ≥ t1 ≥ ... ≥ ti ≥ ti+1 ≥ ti+2 ≥ ... ≥ tj ≥ tj+1 ≥ ... ≥ tn+2 . It follows that n+3 there are 2 4 distinct keychains in Sij . Any fuzzy subgroup µ given by an element of Sij on Cij for any ﬁxed 0 ≤ i0 ≤ j0 ≤ n − 1 is distinct from any fuzzy subgroup on C0 0 ⊂ p ⊂ p2 ⊂ ... ⊂ pn ⊂ pn q since µ is represented by ... ⊂ (pi0 q)ti0 +1 ⊂ ... which cannot appear in any fuzzy subgroup on C0 . For the same reason, µ is distinct from any fuzzy subgroup on Ck 0 ⊂ p ⊂ p2 ⊂ ... ⊂ pk ⊂ pk q ⊂ ... ⊂ pn−1 q ⊂ pr q s ⊂ pn q 2 , where either r = n, s = 1 or r = n − 1, s = 2 and for i0 + 1 < k ≤ n − 1. Now if 1 ≤ k ≤ i0 + 1, then µ is distinct from any fuzzy subgroup ν on Ck since ν would be represented by ... ⊂ (pk−1 q)tk ⊂ ... which cannot appear in µ. We now compare µ with any fuzzy subgroup on Ckl for any i0 < k < l ≤ n − 1. It follows that µ is distinct from any fuzzy subgroup on Ckl since µ is represented by ... ⊂ (pi0 q)ti0 +1 ⊂ ... which cannot appear in any fuzzy subgroup on Ckl . If 0 ≤ k < l ≤ i0 , then µ is distinct from any fuzzy subgroup ν on Ckl since ν has in its representation ... ⊂ (pk−1 q)tk ⊂ ... which cannot appear in µ. Proposition 8.4.9. The number of distinct fuzzy subgroups of G = Zpn ⊕ Zq2 is 2 n 2 2n+2+1 2−r − 1, where n ≥ 2. n−r r r=0

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Proof. By the formula in Theorem 8.4.4 with m = 2, the number of maximal chains of G is equal to 1

ri (2 − i) = 3 + 2 (n − 2) +

i=−1

n (n − 1) , 2

where r−1 = 1; r0 = n−1 and r1 = n(n−1) . This sum reduces to 1+2n+ n(n−1) . 2 2 Now only one such maximal chain C0 goes through pn and it is given by 0 ⊂ p ⊂ p2 ⊂ ... ⊂ pn ⊂ pn q ⊂ pn q 2 . The length of this maximal chain is n + 3. Thus by Proposition 8.2.3, the number of fuzzy subgroups on C0 is 2n+3 − 1. There are 2n maximal chains passing through pn−1 q. Following the notation of representing maximal chains as in the previous Proposition 8.4.7, it follows that each such maximal chain can be uniquely written in the following manner: 0 ⊂ p ⊂ p2 ⊂ ... ⊂ pi ⊂ pi q ⊂ pi+1 q ⊂ ... ⊂ pn−1 q ⊂ pn q ⊂ pn q 2 or 0 ⊂ p ⊂ p2 ⊂ ... ⊂ pi ⊂ pi q ⊂ pi+1 q ⊂ ... ⊂ pn−1 q ⊂ pn−1 q 2 ⊂ pn q 2 . By Lemma 8.4.6, the number of fuzzy subgroups on anyone of the above maximal chains is 2n+2 . Therefore, the total number of fuzzy subgroups on these 2n distinct maximal chains is 2n2n+2 . There are n(n−1) maximal chains passing through pn−1 q 2 . By Lemma 8.4.6, 2 the number of fuzzy subgroups on anyone of these maximal chains is 2n+1 . distinct maximal Thus the total number of fuzzy subgroups on these n(n−1) 2 chains is n (n − 1) n+1 2 . 2 n(n−1) 2n 2 + 2×4 ) which n(n−1) n+1 2 found im2

For m = 2, the formula in Theorem 8.4.5 gives 2n+3 (1 + agrees with the sum of numbers 2n+3 − 1, 2n2n+2 , and mediately above.

Proof of Theorem 8.4.5: The formula in Theorem 8.4.4 gives the numm−1 ber of maximal chains of G to be equal to i=−1 ri (m − i) ∀m ∈ N. It follows that this sum reduces to

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8 Equivalence of Fuzzy Subgroups of Finite Abelian Groups

m n m , where m ≤ n. n−r r r=0 n m For any r = k, 0 ≤ k ≤ m, in the above sum, the term n−k k represents the total number of elements in the set Spn−k qk of all maximal chains passing through pn−k q k . First we notice that the length of any maximal chain is n + m + 1 and therefore the number of fuzzy subgroups on any of the maximal chain is 2n+m+1 − 1. By an argument similar to the one used in the above two lemmas, it follows that 2n+m+1 − 1 + 1 2k many fuzzy subgroups on each of the element of Spn−k qk are distinct from any other fuzzy subgroup on any other maximal chain. Similarly, the number of fuzzy subgroups on other maximal chains passing through pn−r q r , 0 ≤ r ≤ k − 1 can be calculated. As r varies over 0 to m, we get the total number of distinct fuzzy subgroups on G. This total is equal to the sum found in the formula of the theorem.

8.5 Sums, Unions, and Intersections In this section, we study the operations sum, intersection and union, and their behavior with respect to the equivalence ∼ on a group. We examine the extent to which a homomorphism preserves the equivalence. By a ﬂag C on a group G, we mean a chain of subgroups 0 ⊂ G1 ⊂ G2 ⊂ ... ⊂ Gn = G. The Gi ’s are called components of C. We recall that a keychain means 1t1 t2 ...tn , where the ti ’s are not all necessarily distinct. The ti ’s are called pins. The number of distinct ti ’s is called the length of the chain. Two keychains are called distinct if either their lengths diﬀer or one of them contains at least one pin distinct from the other. By a pinned-ﬂag on G, we mean a pair (C, l) of a ﬂag C and a keychain l. Every keychain gives rise to a fuzzy subgroup µ corresponding to a pinned-ﬂag on G given by {0}1 ⊂ (G1 )t1 ⊂ (G2 )t2 ⊂ ... ⊂ (Gn )tn in the following way:  1 if x = 0,      t1 if x ∈ G1 \{0}, µ(x) = t2 if x ∈ G2 \G1 ,   ...    tn if x ∈ Gn \Gn−1 , where the component Gn = G and 1 ≥ t1 ≥ t2 ≥ ... ≥ tn ≥ 0. We denote this simply by (Gn )tn = Gtn . In this case, we say µ is represented by the pinnedﬂag {0}1 ⊂ (G1 )t1 ⊂ (G2 )t2 ⊂ ... ⊂ (Gn )tn . Given a homomorphism between

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two groups, we examine the equivalence classes of homomorphic images and pre-images of fuzzy subgroups. Proposition 8.5.1. Let f : G → H be a homomorphism of a group G into a group H. If µ ∼ ν, then f (µ) ∼ f (ν). Proof. Clearly µ∗ = ν ∗ implies f (µ∗ ) = f (ν ∗ ) which in turn implies f (µ)∗ = f (ν)∗ . Let x, y ∈ H be such that f (µ)(x) > f (µ)(y). It suﬃces to show that f (ν)(x) > f (ν)(y). Now there exists x ∈ G such that f (x ) = x and µ(x ) > µ(y ) for all y ∈ G such that f (y ) = y. Since µ ∼ ν, ν(x ) > ν(y ) for all y ∈ G. Thus f (ν)(x) = ∨{µ(x )|f (x ) = x} > ∨{µ(y )|f (y ) = y} = f (ν)(y). Hence f (µ) ∼ f (ν). Let f : G → H be a homomorphism of a group G into a group H. It follows easily that if µ ∼ ν in H, then f −1 (µ) ∼ f −1 (ν). Also one could consider the behavior of non equivalent fuzzy subgroups under a homomorphism. The following example illustrates that two not equivalent fuzzy subgroups may have equivalent images under a homomorphism. Example 8.5.2. Consider the group Z6 = {0, 1, 2, 3, 4, 5} . Deﬁne f : Z6 → Z6 by f (n) = 2n ∀n ∈ Z6 . Then f is a homomorphism of Z6 into itself. Deﬁne the  fuzzy subgroups µ and ν of Z6 as follows: ∀x ∈ Z6 ,  1 if x = 0. µ(x) = 1/2 if x ∈ {2, 4},   1/4 otherwise; if x = 0, 1 ν(x) = 1/2 if x ∈ {3},  1/4 otherwise. Clearly, µ and ν are not equivalent. Now f (µ)(2) = ∨{µ(x )|f (x ) = 2} = µ(1) ∨ µ(4) = 1/2 and f (µ)(4) = ∨{µ(x )|f (x ) = 4} = µ(2) ∨ µ(5) = 1/2. Also, f (ν)(2) = ∨{ν(x )|f (x ) = 2} = ν(1) ∨ ν(4) = 1/4 and f (ν)(4) = = 4} = µ(2) ∨ µ(5) = 1/4. Hence it follows that ∨{µ(x )|f (x )  if x = 0 1 f (µ)(x) = 1/2 if x ∈ {2, 4}  if x ∈ {1, 3, 5} 0 if x = 0, 1 f (ν)(x) = 1/4 if x ∈ {2, 4}  0 if x ∈ {1, 3, 5} Clearly, f (µ) and f (ν) are equivalent. Similarly, we may have non equivalent fuzzy subgroups giving rise to equivalent pre-images under a homomorphism. This can be seen by the following example.

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8 Equivalence of Fuzzy Subgroups of Finite Abelian Groups

Example 8.5.3. Let Z6 and f be as in the previous example. Deﬁne the fuzzy subgroups  µ and ν of Z6 as follows: ∀x ∈ Z6 ,  1 if x = 0. µ(x) = 1/2 if x ∈ {2, 4},   1/3 otherwise; if x = 0, 1 ν(x) = 1/2 if x ∈ {3},  1/3 otherwise. Clearly µ is not equivalent to ν. Using the equations f −1 (µ)(x) = µ(f (x)) and f −1 (ν)(x) =ν(f (x)), it follows that 1 if x ∈ {0, 3}, f −1 (µ)(x) = 1/2 otherwise; 1 if x ∈ {0, 3}, f −1 (ν)(x) = 1/3 otherwise. Thus, f −1 (µ) and f −1 (ν) are equivalent. If f : G → H is an epimorphism, then f (f −1 (µ)) = µ. Thus if µ and ν are not equivalent fuzzy subgroups of H, then f −1 (µ) and f −1 (ν) are not equivalent fuzzy subgroups of G. However, if f : G → H is a monomorphism, a similar conclusion cannot be drawn. For example, let G = Z2 = {0, 1} and H = {0 }. Deﬁne f : G → H by f (0) = 0 = f (1). Deﬁne µ : G → [0, 1] by µ(0) = 1 and µ(1) = 1/2. Then f −1 (f (µ))(1) = f (µ)(f (1)) = f (µ)(0 ) = µ(0) ∨ µ(1) = 1 = 1/2 = µ(1). Thus f −1 (f (µ)) = µ. Deﬁne ν : G → [0, 1] by ν(0) = 1 = ν(1). Then µ ν. However, f (µ)(0 ) = µ(0) ∨ µ(1) = 1 = ν(0) ∨ ν(1) = f (ν)(0 ). thus f (µ) ∼ f (ν). In general the operations of intersection, union and sum of fuzzy subgroups do not preserve the equivalence classes of fuzzy subgroups. This is shown in the following example. Example 8.5.4. Consider the group of integers Z under addition. Deﬁne the fuzzy subgroups µ, µ , ν, and ν of Z as follows: ∀x ∈ Z, 1 if x ∈ 2Z, 1 if x ∈ 2Z, µ(x) = µ (x) = 1/4 otherwise 1/3 otherwise 1 if x ∈ 3Z, 1 if x ∈ 3Z, ν(x) = ν (x) = 1/2 otherwise 1/5 otherwise. Then,

 if x ∈ 6Z, 1 (µ ∩ ν)(x) = 1/2 if x ∈ 2Z\6Z,  1/3 otherwise; if x ∈ 6Z, 1 (µ ∩ ν )(x) = 1/4 if x ∈ 3Z\6Z,  1/5 otherwise.

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Clearly, µ ∼ µ and ν ∼ ν . However, it follows that (µ ∩ ν) (µ ∩ ν ) since (µ ∩ ν)(3) = 13 < (µ ∩ ν)(2) = 12 , but (µ ∩ ν )(3) = 14 > (µ ∩ ν )(2) = 15 . Similarly, it can be shown that µ ∼ µ and ν ∼ ν does not imply µ ∪ ν ∼ µ ∪ ν.

The next example deals with the operation, sum. Example 8.5.5. Consider the group of integers Z under addition. Deﬁne the fuzzy subgroups  ∈ Z,  µ, µ , ν, and ν of Z as follows: ∀x if x ∈ 6Z, if x ∈ 6Z, 1 1 µ (x) = 3/4 if x ∈ 2Z\6Z, µ(x) = 1/2 if x ∈ 2Z\6Z,    1/4 otherwise,  1/8 otherwise if x ∈ 6Z, if x ∈ 6Z, 1 1 ν(x) = 2/3 if x ∈ 3Z\6Z, ν (x) = 1/2 if x ∈ 3Z\6Z,   1/6 otherwise, 1/6 otherwise. Now (µ + ν)(x) = ∨{µ(y) ∧ ν(z)|x = y + z} for all x ∈ Z and similarly for µ + ν . Thus,   1 if x ∈ 6Z, (µ + ν)(x) = 2/3 if x ∈ 3Z\6Z,  1/2 otherwise,  1 if x ∈ 6Z,  (µ + ν )(x) = 3/4 if x ∈ 2Z\6Z,  1/2 otherwise. Clearly, µ ∼ µ and ν ∼ ν . However, (µ + ν) (µ + ν ) since (µ + ν)(3) = 23 > (µ + ν)(2) = 12 , but (µ + ν )(3) = 12 < (µ + ν )(2) = 34 . In the previous example, µ + ν ∼ ν. However, this need not be true in general. For example if 16 is replaced by 0 in the deﬁnition of ν above, then µ + ν ν since the support of µ + ν is now diﬀerent than that of ν. Also we note, µ + ν µ. Thus µ + ν is not equivalent to either µ or ν. In Proposition 8.5.8, we show that if µ ∼ ν, then µ + ν ∼ µ and so µ + ν ∼ ν. If we take two fuzzy subgroups µ and ν from the same equivalence class C determined by µ and ν, then the inf, sup and sum of µ and ν determine the same equivalence class C. Proposition 8.5.6. Let µ and ν be fuzzy subgroups of a group G. If µ ∼ ν, then µ ∩ ν ∼ µ. Proof. Since µ ∼ ν, µ∗ = ν ∗ . Since also (µ ∩ v)∗ = µ∗ ∩ ν ∗ , we have (µ ∩ ν)∗ = µ∗ . Let x, y ∈ G. If µ(x) > µ(y), then ν(x) > ν(y) and so (µ ∩ ν)(x) > (µ ∩ ν)(y). Now suppose (µ ∩ ν)(x) > (µ ∩ ν)(y). We have four cases. (1) µ(x) = (µ ∩ ν)(x) and µ(y) = (µ ∩ ν)(y) : In this case, clearly µ(x) > µ(y).

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8 Equivalence of Fuzzy Subgroups of Finite Abelian Groups

(2) µ(x) = (µ ∩ ν)(x) and ν(y) = (µ ∩ ν)(y) : Here ν(x) ≥ µ(x) > ν(y). Thus µ(x) > µ(y) since µ ∼ ν. (3) ν(x) = (µ ∩ ν)(x) and ν(y) = (µ ∩ ν)(y) : Here ν(x) > ν(y). Thus µ(x) > µ(y) since µ ∼ ν. (4) ν(x) = (µ ∩ ν)(x) and µ(y) = (µ ∩ ν)(y) : In this case, ν(x) > µ(y). Since µ(x) ≥ ν(x), µ(x) > µ(y). Proposition 8.5.7. Let µ and ν be fuzzy subgroups of a group G. If µ ∼ ν then, µ ∪ ν ∼ µ. Proof. (µ ∪ ν)∗ = µ∗ ∪ ν ∗ = µ∗ . Let x, y ∈ G. Suppose (µ ∪ ν)(x) > (µ ∪ ν)(y). Suppose µ(x) = (µ ∪ ν)(x). Then µ(x) > µ(y) ∨ ν(y). Thus µ(x) > µ(y). Suppose ν(x) = (µ ∪ ν)(x). Then ν(x) > µ(y) ∨ ν(y) and so ν(x) > ν(y). Since µ ∼ ν, µ(x) > µ(y). Suppose µ(x) > µ(y). Then ν(x) > ν(y) since µ ∼ ν. Thus µ(x) ∨ ν(x) > µ(y) ∨ ν(y). Hence (µ ∪ ν)(x) > (µ ∪ ν)(y). Proposition 8.5.8. Let µ and ν be fuzzy subgroups of a ﬁnite group G. If µ ∼ ν, then µ + ν ∼ µ. Proof. (µ + ν)∗ = µ∗ + ν ∗ = µ∗ . Suppose (µ + ν)(x) > (µ + ν)(y). Then there are x1 and x2 with x = x1 + x2 such that µ(x1 ) ∧ ν(x2 ) > (µ + ν)(y). Now (µ + ν)(y) ≥ µ(y) ∧ ν(e) = µ(y). Suppose µ(x1 ) = µ(x1 ) ∧ ν(x2 ). Then µ(x1 ) > µ(y) and ν(x2 ) ≥ µ(x1 ) > ν(y). Thus µ(x2 ) > µ(y) since µ ∼ ν. Therefore, µ(x) ≥ µ(x1 ) ∧ µ(x2 ) > µ(y). Suppose ν(x2 ) = µ(x1 ) ∧ ν(x2 ). Then µ(x1 ) ≥ ν(x2 ) > ν(y). Thus µ(x2 ) > µ(y) since µ ∼ ν. Also µ(x1 ) ≥ ν(x2 ) > (µ + ν)(y) ≥ µ(y) ∧ ν(e) = µ(y). Hence µ(x) > µ(y). Conversely, suppose µ(x) > µ(y). Then we show by contradiction that (µ + ν)(x) > (µ + ν)(y). Suppose (µ + ν)(x) < (µ + ν)(y). Then by the above argument, we have µ(y) > µ(x), a contradiction. Suppose (µ + ν)(x) = (µ + ν)(y). Then since G is ﬁnite, there exists y1 , y2 ∈ G such that (µ + ν)(x) = µ(y1 ) ∧ ν(y2 ) and y = y1 + y2 . Hence µ(y1 ) ≥ (µ + ν)(x) ≥ µ(x). Also, ν(y2 ) ≥ ν(x) and so µ(y2 ) ≥ µ(x) since µ ∼ ν. Thus µ(y) ≥ µ(y1 ) ∧ µ(y2 ) ≥ µ(x), a contradiction. Consequently, (µ + ν)(x) > (µ + ν)(y). Proposition 8.5.9. Let µ and ν be fuzzy subgroups of a group G. If µ ∼ ν, then 1 − µ ∼ 1 − ν. Proof. Now x ∈ (1 − µ)∗ ⇔ (1 − µ)(x) > 0 ⇔ 0 ≤ µ(x) < 1 ⇔ 0 ≤ ν(x) < 1 ⇔ (1 − ν)(x) > 0 ⇔ x ∈ (1 − ν)∗ . Thus (1 − µ)∗ = (1 − ν)∗ . Let x, y ∈ G. Suppose (1 − µ)(x) > (1 − µ)(y). Then µ(x) < µ(y). Thus ν(x) < ν(y). Hence (1 − ν)(x) > (1 − ν)(y). Similarly, (1 − ν)(x) > (1 − ν)(y) implies (1 − µ)(x) > (1 − µ)(y). We now determine the equivalence class of fuzzy subgroups corresponding to the intersection and sum of two fuzzy subgroups in terms of pinned-ﬂags associated with given fuzzy subgroups. Throughout this section, we require the number of components in a pinned-ﬂag to be at least 3, otherwise the discussion becomes trivial. Therefore, we assume n ≥ 2.

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In the next proposition, we look at a special case, namely the characterization of intersection and sum of two equivalence classes of fuzzy subgroups with the same ﬂag of subgroups as their t-cuts. Proposition 8.5.10. Let G be a group with a ﬂag C of subgroups {0} = G0 ⊂ G1 ⊂ G2 ⊂ ... ⊂ Gn = G. Suppose µ = (Cµ , lµ ) and ν = (Cν , lν ) are two fuzzy subgroups whose representative keychains are of the form lµ = 1t1 t2 ...tn and lν = 1s1 s2 · · · sn , respectively. Then the following assertions hold: u ∩ ν ∼ 1t1 ∧ s1 ...tn ∧ sn , (1) Cµ∩ν = C u + ν ∼ 1t1 ∨ s1 ...tn ∨ sn . (2) Cµ+ν = C Proof. Let x ∈ G. Then there is an index i such that 1 ≤ i ≤ n with x ∈ Gi , but x ∈ / Gi−1 . Thus µ(x) = ti and ν(x) = si . (1) It suﬃces to prove that (µ ∩ ν)(x) = ti ∧ si . However, this is clearly true. (2) We have that (µ + ν)(x) ≥ ti ∨ si . Suppose (µ + ν)(x) > ti ∨ si . Then there exist x1 , x2 ∈ G such that x = x1 + x2 and µ(x1 ) ∧ ν(x2 ) > ti ∨ si . Hence µ (x1 ) > ti and ν(x2 ) > si and so x1 ∈ Gi−1 and x2 ∈ Gi−1 . Thus x ∈ Gi−1, a contradiction. In the next proposition, we consider two ﬂags diﬀering in one component. Proposition 8.5.11. Suppose µ and ν are fuzzy subgroups of G whose representative keychains are of the form 1t1 t2 · · · tn and whose underlying ﬂags are Cµ and Cν described as follows: Cµ : {0} = G0 ⊂ G1 ⊂ G2 ⊂ ... ⊂ Gk−1 ⊂ Gk ⊂ Gk+1 ⊂ ... ⊂ Gn = G and Cν : {0} = G0 ⊂ G1 ⊂ G2 ⊂ ... ⊂ Gk−1 ⊂ Hk ⊂ Gk+1 ⊂ ... ⊂ Gn = G for a ﬁxed k such that 1 ≤ k ≤ n − 1, Gk = Hk . Then the following assertions hold: (1) µ ∩ ν = (Cµ∩ν , lµ∩ν ) : 1t1 t2 ...tk−1 tk+1 tk+1 ...tn on Cµ or on Cν . and (2) µ + ν = (Cµ+ν , lµ+ν ) :1t1 t2 ...tk−1 tk tk tk+2 ...tn on Cµ or on Cν , where Cµ∩ν = Cµ = Cν and Cµ+ν = Cµ = Cν . Proof. (1) By Proposition 8.5.10, it follows that (µ ∩ ν)(x) has the same keychain pins as µ and ν on all Gi ’s , for i = 1, 2, ..., k −1, k +1, ..., n. It suﬃces to prove the case for i = k. Let x ∈ Gk+1 \Gk−1 . Then (µ ∩ ν)(x) ≥ tk+1 . Suppose (µ ∩ ν)(x) > tk+1. Then either (µ ∩ ν)(x) = ts for some 1 ≤ s ≤ k or (µ ∩ ν)(x) = 1. The latter case implies µ(x) = 1 = ν(x). However, x ∈ / Gk−1 implies µ(x) < tk−1 . This in turn implies µ(x) = 1, a contradiction. For the former case, ts ≥ tk . Thus µ (x) ≥ tk . Hence x ∈ Gk . Similarly, x ∈ Hk . Thus x ∈ Gk ∩ Hk . By the maximality of the chains involved, Gk ∩ Hk = Gk−1 . Therefore, x ∈ Gk−1 , which is a contradiction of the choice of x. Thus (µ ∩ ν)(x) = tk+1 for all x ∈ Gk+1 \Gk−1 .

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(2) It follows that Gk+1 = Gk + Hk by the maximality of chains. Similar to the proof of case (1), for the sum, it suﬃces to consider x ∈ Gk+1 \Gk−1 . Then (µ ∩ ν)(x) ≥ µ(x1 ) ∧ ν(x2 ) ≥ tk for x = x1 + x2 with x1 ∈ Gk and x2 ∈ Hk . Suppose µ(x1 ) ∧ ν(x2 ) > tk . Then either µ(x1 ) ∧ ν(x2 ) = ts for some 1 ≤ s ≤ k − 1 or µ (x1 ) ∧ ν (x2 ) = 1. The latter case implies x = x1 + x2 ∈ Gk−1 + Gk−1 = Gk−1 , and in the former case x = x1 + x2 ∈ Gs + Gs = Gs . By the chain property, we conclude that x ∈ Gk−1 , which is a contradiction of the choice of x. Thus (µ ∩ ν)(x) = tk for all x ∈ Gk+1 \Gk−1 . Let µ and ν be fuzzy subgroups of a group whose pinned-ﬂags diﬀer in two nonconsecutive components and are given by µ = (Cµ , lµ ) and ν = (Cν , lν ) as follows: Cµ : {0} = G0 ⊂ G1 ⊂ ... ⊂ Gi1 ⊂ ... ⊂ Gi2 ⊂ ... ⊂ Gik ⊂ ... ⊂ Gn = G and Cν : {0} = G0 ⊂ G1 ⊂ ... ⊂ Hi1 ⊂ ... ⊂ Hi2 ⊂ ... ⊂ Hik ⊂ ... ⊂ Gn = G, where 1 ≤ i1 < i1 + 2 ≤ i2 < i2 + 2 ≤ ... < ik−1 < ik−1 + 2 ≤ ik ≤ n − 1, Gik = Hki for k = 1, 2. Then, using a similar argument as in Proposition 8.5.11 inductively, we have the following result. Corollary 8.5.12. Let µ = (Cµ , lµ ) and ν = (Cν , lν ) be as above. Then the following assertions hold: (1) µ ∩ ν is represented by the keychain lµ∩ν = 1t1 ...ti1 −1 ti1 +1 ti1 +1 ...tik −1 tik +1 tik +1 ...tn on Cµ∩ν = Cµ = Cν (2) µ + ν is represented by the keychain lµ+ν = 1t1 ...ti1 −1 ti1 ti1 ...tik −1 tik tik ...tn on Cµ+ν = Cµ = Cν . We now consider two ﬂags diﬀering in two or more components consecutively, with all other basic assumptions taken to be true on the ﬂags as in the previous propositions: Cµ : ... ⊂ Gi−1 ⊂ Gi ⊂ Gi+1 ⊂ ... ⊂ Gi+k−1 ⊂ Gi+k ⊂ Gi+k+1 ⊂ ... and Cν : ... ⊂ Gi−1 ⊂ Hi ⊂ Hi+1 ⊂ ... ⊂ Hi+k−1 ⊂ Hi+k ⊂ Gi+k+1 ⊂ ... for 1 ≤ i < ... < i + k ≤ n − 1,with k ≥ 1 and Gi+k = Hi+k . In the above, we have only indicated the corresponding distinct components in Cµ and Cν as the suppressed corresponding components are assumed to be identical in the two ﬂags. Cµ∩ν : ... ⊂ Gi−1 ⊂ Gi+1 ∩ Hi+1 ⊂ ... ⊂ Gi+k ∩ Hi+k ⊂ F ⊂ Gi+k+1 ⊂ ..., where F can be either Gi+k or Hi+k . Cµ+ν : ... ⊂ Gi−1 ⊂ E ⊂ Gi + Hi ⊂ ... ⊂ Gi+k−1 + Hi+k−1 ⊂ Gi+k+1 ⊂ ..., where E can be either Gi or Hi . In the above, we have only indicated the corresponding distinct components in Cµ , Cν , Cµ∩ν and Cµ+ν , and as such the suppressed corresponding components are assumed to be identical in the two ﬂags.

8.5 Sums, Unions, and Intersections

233

Proposition 8.5.13. Suppose µ and ν are fuzzy subgroups of a group G whose representative keychains are of the form 1t1 t2 ...tn and whose underlying ﬂags are Cµ and Cν , respectively, as described above. Then the following assertions hold: (1) µ ∩ ν is represented by the keychain 1t1 t2 ...ti−1 ti+1 ti+2 ...ti+k ti+k+1 ti+k+1 ti+k+2 ...tn on Cµ∩ν : ... ⊂ Gi−1 ⊂ Gi+1 ∩ Hi+1 ⊂ ... ⊂ Gi+k ∩ Hi+k ⊂ F ⊂ Gi+k+1 ⊂ ..., where F can be either Gi+k or Hi+k and (2) µ + ν is represented by the keychain 1t1 t2 ...ti−1 ti ti ti+1 ...ti+k ti+k+2 ...tn on Cµ+ν : ... ⊂ Gi−1 ⊂ E ⊂ Gi + Hi ⊂ ... ⊂ Gi+k−1 + Hi+k−1 ⊂ Gi+k+1 ⊂ ..., where E can be either Gi or Hi . Proof. As in Proposition 8.5.11, it suﬃces to consider only indices i, i+1, ..., i+ k. (1) Let x ∈ Gi+1 ∩ Hi+1 \ Gi−1 . Then clearly ti−1 > (µ ∩ ν)(x) ≥ ti+1 . Now suppose (µ ∩ ν)(x) > ti+1. Then (µ ∩ ν)(x) = ti . Thus x ∈ Gi ∩ Hi = Gi−1 by the maximality of Cµ and Cν . This is a contradiction of our choice of x. Hence (µ ∩ ν)(x) = ti+1 . Now for other cases, let x ∈ Gi+j ∩ Hi+j \ Gi+j−1 ∩ Hi+j−1 for j = 2, 3, ..., k. Then ti+j−1 > (µ ∩ ν)(x) ≥ ti+j . Suppose (µ∩ν)(x) > ti+j . Then there exists a pin ts representing the value of (µ∩ν)(x) and is such that ti+j−1 > ts > ti+j , However, this is a contradiction since ti+j−1 and ti+j are two consecutive pins. Thus (µ ∩ ν)(x) = ti+j . Let x ∈ F \ Gi+k ∩ Hi+k , say F = Hi+k . Then ν(x) ≥ ti+k and µ(x) < ti+k which implies (µ ∩ ν)(x) = µ(x) < ti+k . However, (µ ∩ ν)(x) ≥ ti+k+1 . Suppose (µ ∩ ν)(x) > ti+k+1 . Then this leads to a contradiction as in the previous case. Thus (µ ∩ ν)(x) = ti+k+1 . The argument is similar for F = Gi+k . Finally, let x ∈ Gi+k+1 \F , say F = Hi+k . Then (µ ∩ ν)(x) ≥ ti+k+1 and ν(x) < ti+k which implies (µ ∩ ν)(x) < ti+k . As in the previous cases, it follows that (µ ∩ ν)(x) = ti+k+1 . The argument is similar for F = Gi+k Thus (1) is proved. (2) Let x ∈ E\Gi−1 , say E = Hi . Then ν(x) ≥ ti . Hence (µ ∩ ν)(x) ≥ ti . Suppose (µ ∩ ν)(x) ≥ ti−1. Then x ∈ Gi−1 , a contradiction. Thus ti ≤ (µ ∩ ν)(x) < ti−1. As in case (1) above, we conclude that (µ ∩ ν)(x) = ti. Next let x ∈ Gi + Hi \ E, say E = Hi. Then (µ ∩ ν)(x) ≥ ti and ν(x) < ti . Suppose (µ ∩ ν)(x) ≥ ti−1 . Then x ∈ Gi−1 ⊆ Hi , a contradiction. Thus ti ≤ (µ ∩ ν)(x) < ti−1 . Hence, as before (µ ∩ ν)(x) = ti . The argument is

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similar for E = Gi . For other cases, let x ∈ Gi+j + Hi+j \ Gi+j−1 + Hi+j−1 , for j = 1, 2, ..., k − 1. Then ti+j−1 > (µ ∩ ν)(x) ≥ ti+j . As in other parts of this proof, it is clear that (µ ∩ ν)(x) = ti+j . Finally, let x ∈ Gi+k+1 \ Gi+k−1 +Hi+k−1 . Then (µ∩ν)(x) ≥ ti+k+1 . However, Gi+k+1 = Gi+k +Hi+k , by the maximality of the chains Cµ and Cν . Hence ti+k−1 > (µ ∩ ν)(x) ≥ ti+k . Thus (µ ∩ ν)(x) = ti+k . The determination of the pinned-ﬂags of the intersection and the sum of two fuzzy subgroups µ and ν, where the pins as well as the ﬂags of the pinnedﬂags Cµ and Cv representing µ and ν are distinct, in general, does not seem to follow any particular pattern as we have derived above. This is illustrated by the following example. Example 8.5.14. Let G = Z72. Let Cµ and Cv be the pinned-ﬂags of µ and ν on G given by 1 (Cµ , lµ ) : 01 ⊂ (Z3 ) 12 ⊂ (Z9 ) 15 ⊂ (Z18 ) 16 ⊂ (Z36 ) 19 ⊂ (Z72 ) 10

and 1 (Cν , lν ) : 01 ⊂ (Z3 ) 13 ⊂ (Z6 ) 14 ⊂ (Z12 ) 17 ⊂ (Z36 ) 18 ⊂ (Z72 ) 11

respectively. It follows that the pinned-ﬂags for µ ∩ ν and µ + ν are 1 (Cµ∩ν , lµ∩ν ) : 01 ⊂ (Z3 ) 13 ⊂ (Z6 ) 16 ⊂ (Z18 ) 18 ⊂ (Z36 ) 19 ⊂ (Z72 ) 11

and 1 , (Cµ+ν , lµ+ν ) : 01 ⊂ (Z3 ) 12 ⊂ (Z6 ) 14 ⊂ (Z18 ) 15 ⊂ (Z36 ) 17 ⊂ (Z72 ) 10

respectively. In the above calculation, the roles played by the pins and the components of the ﬂags are equally important in a way in which they are tied to each other. Suppose we retain the ﬂags, but the pins for µ and ν are changed 1 1 1 1 1 1 1 1 to 1 13 19 18 36 72 and 1 72 90 100 110 120 respectively. Then µ ∩ ν and µ + ν have the pinned-ﬂags given by 1 ⊂ (Z6 ) 1 ⊂ (Z12 ) 1 1 1 (Cµ ∩ν , lµ ∩ν ) : 01 ⊂ (Z3 ) 72 ⊂ (Z36 ) 110 ⊂ (Z72 ) 120 90 100 1 ⊂ (Z36 ) 1 ⊂ (Z72 ) 1 , (Cµ +ν , lµ +ν ) : 01 ⊂ (Z3 ) 13 ⊂ (Z9 ) 19 ⊂ (Z18 ) 18 36 72

respectively. Similarly, we could retain the pins, but change the ﬂags of µ and ν, for example in (1) and (2) above we retain the same pins, but swap the underlying ﬂags. Then it follows that we arrive at diﬀerent (from (3) and (4)) pinned-ﬂags for µ ∩ ν and µ + ν.

8.6 Fuzzy Subgroups of Inﬁnite Cyclic Groups

235

8.6 Fuzzy Subgroups of Inﬁnite Cyclic Groups One of the main purposes of crisp group theory is to determine the number of distinct subgroups a given group possesses. The number of subgroups of an inﬁnite group is inﬁnite and so to characterize distinct subgroups we use the concepts of homomorphism and isomorphism. The determination of how many distinct fuzzy subgroups a given group can possess is an interesting area of research within fuzzy group theory. Several authors have studied the issue of how many distinct fuzzy subgroups a group may have. They have used various concepts including those of fuzzy isomorphism [6, 7] and the cardinality of the set of truth-values [18]. The concept of fuzzy isomorphism incorporates the following notions: Let G be a group and µ and ν be fuzzy subgroups of G. If there exists an isomorphism f : µ∗ → ν ∗ such that µ(x) > µ(y) implies ν(f (x)) > ν(f (y))∀x, y ∈ µ∗ , then µ is said to be isomorphic to ν. In Section 8.1, the concepts of fuzzy isomorphism [6, 7] and fuzzy equivalence were compared and it was observed that equivalence is ﬁner than isomorphism. Hence the number of fuzzy subgroups of certain ﬁnite abelian groups was studied using the equivalence of fuzzy subsets. However, the equivalence approach pays less attention to the key point in group theory that two groups are considered to be the same group if their algebraic properties are same, i.e., they are isomorphic. Since the number of crisp subgroups of a ﬁnite group G is also ﬁnite, it is possible from the point of view of equivalence to count the number of fuzzy subgroups of G. Thus it seems that in some special cases, equivalence does work. However, for inﬁnite groups, for example an inﬁnite cyclic group, we can construct inﬁnite fuzzy subgroups from the equivalence viewpoint as indicated in Example 8.6.2 below. Thus it seems that in the case of inﬁnite groups, the equivalence relation is not suitable. Similar problems are encountered when using fuzzy isomorphisms. For example, let G = a be an inﬁnite cyclic group and µ and ν be fuzzy subgroups of G such that µ∗ = ν ∗ = G. Then an isomorphism f of µ∗ onto ν ∗ is either the identity or is such that f (x) = −x∀x ∈ G. In this case, µ and ν are equivalent if and only if they are isomorphic. Hence a new approach is needed to study the classiﬁcation of fuzzy subgroups of an inﬁnite group. We use the concept of group isomorphism to characterize the similarity of fuzzy subgroups. Deﬁnition 8.6.1. Let µ and ν be fuzzy subgroups of a group G. If |Im(µ)| = |Im(ν)|, µ∗ ∼ = ν ∗ , and for all t ∈ [0, 1] such that µt = ∅ there exists s ∈ [0, 1] such that µt ∼ = νs and for all s ∈ [0, 1] such that νs = ∅ there exists t ∈ [0, 1] such that µt ∼ = νs , then µ and ν are called S ∗ -equivalent, written µ ∼ = ν. If in Deﬁnition 8.6.1, µ∗ ∼ = ν ∗ and µt ∼ = νs is replaced by µ∗ = ν ∗ and ∗ µt = νs , respectively, then the notion of S -equivalence becomes that of strong equivalence given in Deﬁnition 8.1.10. If two fuzzy subgroups of a ﬁnite group are equivalent, then they are S ∗ -equivalent. However the converse is not necessarily true since two subgroups of a group may be isomorphic without being

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equal. Thus the notion of equivalence is a special case of S ∗ -equivalence when dealing with fuzzy subgroups of ﬁnite groups. The initial idea of Deﬁnition 8.6.1 ﬁrst appeared in [16]. Example 8.6.2. Let G be an inﬁnite cyclic group generated by a. Let µ and ν be fuzzy subsets of G deﬁned as follows. ∀x ∈ G, µ(x) = 1 if x ∈ a2 , µ(x) = 1/2 otherwise and ν(x) = 1 if x ∈ a3 and ν(x) = 1/2 otherwise. Then µ and ν are fuzzy subgroups of G. Clearly µ and ν are neither equivalent nor isomorphic, but they are S ∗ -equivalent. Example 8.6.3. Let G be an inﬁnite cyclic group generated by a. Let {tn }∞ n−1 be a strictly increasing sequence of numbers in [0, 1]. Let µ and ν be the fuzzy subsets of G deﬁned as follows: ∀x ∈ G, µ(x) = t1 if x ∈ G\a2 , µ(x) = tn if x ∈ a2n−2 \a2n for n = 2, 3, ..., µ(e) = 1 and ν(x) = t1 if x ∈ G\a3 , ν(x) = tn if x ∈ a3n−3 \a3n for n = 2, 3, ..., ν(e) = 1. Then µ and ν are fuzzy subgroups of G which are neither equivalent nor isomorphic. It is shown in Theorem 8.6.6 that µ and ν are S ∗ -equivalent. Proposition 8.6.4. Let G be an inﬁnite cyclic group generated by a. Let µ and ν be ﬁnite-valued fuzzy subgroups of G. Then µ ∼ = ν if and only if |Im(µ)| = |Im(ν)|, µ∗ ∼ = ν ∗ , and µ∗ ∼ = ν∗ . Proof. Suppose that the conditions hold. Clearly, if t = 0, then µ0 = ν0 = G. Suppose t ∈ (0, 1] and µt = ∅. Suppose µt = {e}. Let s = ν(e). Then µt = µ∗ ∼ = ν∗ = νs . Suppose µt = {e}. Then µt = ak and µ(ak ) ≥ t > 0 for some positive integer k. Since µ∗ ∼ = ν ∗ and µ and ν are ﬁnite-valued, ν ∗ = aj for some positive integer j and s = ∧{ν(x) | ν(x) > 0} > 0. Then νs = aj and µt ∼ = νs . Conversely, suppose µ ∼ = ν. Suppose µ∗ = {e}. There is an s ∈ [0, 1] such that µ∗ ∼ = νs . Thus νs = {e}.Since ν∗ ⊆ νs and s ≤ ν(e), it follows that ν∗ = νs . Thus µ∗ ∼ = ν∗ . Suppose µ∗ = {e}. Then µ∗ = ak for some positive integer k. there exists t ∈ [0, 1] such that µt ∼ = νν(e) = ν∗ . Thus ν∗ = {e}. Hence ν∗ = aj for some positive integer j. Thus µ∗ ∼ = ν∗ . Lemma 8.6.5. Let G be an inﬁnite cyclic group generated by a. Let µ be a fuzzy subgroup of G. If Im(µ) is an inﬁnite set, then µ∗ = {e}. Proof. Suppose µ∗ = {e}. Then µ∗ = ak for some positive integer k. Since ∀t ≤ µ(e), µ∗ ⊆ µt , there is a positive integer j such that µt = aj and 1 ≤ j ≤ k. Thus the number of level subgroups is ﬁnite and so Im(µ) is ﬁnite, a contradiction. Hence µ∗ = {e}. Theorem 8.6.6. Let G be an inﬁnite cyclic group generated by a. Let µ and ν be fuzzy subgroups of G. If both Im(µ) and Im(ν) are inﬁnite, then µ ∼ = ν. Proof. Since both Im(µ) and Im(ν) are inﬁnite, both Im(µ) and Im(ν) are countable. Thus |Im(µ)| = |Im(ν)|. Furthermore, there exists x ∈ G such that

References

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∼ ν ∗ . For µ(e) > µ(x) > 0. Hence µ∗ = {e}. Similarly, ν ∗ = {e}. Thus µ∗ = all t ∈ [0, 1] such that µt = ∅, either µt = {e} or µt ∼ = G. By Lemma 8.6.5, we have in either case that there is an s ∈ [0, 1] such that µt ∼ = νs . Hence µ∼ = ν. From Theorem 8.6.6, we see that, with respect to S ∗ -equivalence, inﬁnitevalued fuzzy subgroups of an inﬁnite cyclic group are unique. With respect to equivalence, the number of inﬁnite-valued fuzzy subgroups of an inﬁnite cyclic group is inﬁnite. Clearly, the notion of S ∗ -equivalence can be used to reduce the number of distinct fuzzy subgroups of a ﬁnite group compared with the equivalence approach since two fuzzy subgroups which are not equivalent may be S ∗ -equivalent. Remark 8.6.7. The number of fuzzy subgroups of a ﬁnite Abelian group G with respect to a suitable equivalence relation were determined for special cases of G. However, it remains an open problem to determine the number for an arbitrary ﬁnite Abelian group.

References 1. Y. Alkhamees, Fuzzy cyclic subgroups and fuzzy cyclic p-subgroups, J. Fuzzy Math. 3(4) (1995) 911-919. 2. P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264-269. 3. J. B. Fraleigh, A First Course in Abstract Algebra, Addison-Wesley, London (1982). 4. J. Jiashang, C. Degang,Wu Congxin, and E. C. C. Tsang, Some notes on the equivalence of fuzzy sets and fuzzy subgroups, Fuzzy Sets and Systems, to appear. 201 5. B.B Makamba, Studies in Fuzzy Groups, Thesis, Rhodes University, Grahamstown (1992). 6. M. Mashinchi and M. Mukaidono, A classiﬁcation of fuzzy subgroups, Ninth Fuzzy System Symposium, Sapporo, Japan, 1992, 649-652. 235 7. M. Mashinchi and M. Mukaidono, On fuzzy subgroup classiﬁcation, Research Report of Meiji University, Japan, vol. 9 (65) (1993) 31-36. 235 8. M. Mashinchi and Sh. Salili, On fuzzy isomorphism theorems, J. Fuzzy Math. 4 (1996) 39-49. 9. V. Murali and B.B. Makamba, On an Equivalence of Fuzzy Subgroups I, Fuzzy Sets and Systems 123 (2001) 259-264. 201 10. V. Murali and B.B. Makamba, On an Equivalence of Fuzzy Subgroups II, Fuzzy Sets and Systems 136 (2003) 93-104. 201, 208 11. V. Murali and B. B. Makamba, Operations on equivalent fuzzy subgroups, J. Fuzzy Math., to appear. 201 12. V. Murali and B. B. Makamba, Counting the number of fuzzy subgroups of an Abelian group of order pn q m , Fuzzy Sets and Systems 144 (2004) 459-470. 201 13. N. P. Mukherjee and P. Bhattacharya, Fuzzy Groups: Some Group Theoretic Analogs, Inform. Sci. 39 (1986) 247-268.

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14. S. Ray, Isomorphic fuzzy groups, Fuzzy Sets and Systems 50 (2) (1992) 201-207. 15. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. 16. Su-Yun Li, De-Gang Chen,Wen-Xiang Gu, and Hui Wang, Fuzzy homomorphisms, Fuzzy Sets and Systems 79 (1996) 235-238. 236 17. S. Tung and Y. Zhang, Fuzzy subgroups of the same type, Fuzzy Sets and Systems 87 (1997) 377 - 381. 18. S. A. Ziaie, H. V. Kumbhojkar, and M. Mashinchi, On cardinality of truth-value set of a fuzzy subgroup, Int. J. Scientia Iranica 3 (1996) 97-103. 235

9 Lattices of Fuzzy Subgroups

Many results concerning relationships between classes of crisp subsets can be carried over to similar relationships between classes of fuzzy subsets. The purpose of the next few sections is to provide a framework for transferring crisp results into fuzzy results when such a transference is possible. This is accomplished by the presentation of a suitable metatheorem. The results of the ﬁrst four sections of this chapter are mainly from [15, 16].

9.1 Embedding of Fuzzy Power Sets Let X be a non-empty set. Recall that C(X) = {f | f : X → {0, 1}} and that P(X) denotes the power set of X. We call C(X) the crisp power set of X. Then P(X) and C(X) are in a natural one-to-one correspondence and C(X) ⊆ FP(X). In Section 9.4, we present a metatheorem and two closely associated subdirect product representation theorems. Let C(X)J denote {f | f : J → C(X)}, where J is the half open interval J = [0, 1). These theorems arise from ﬁrst establishing a representation function R : FP(X) → C(X)J . The idea behind R is very closely related to the concepts of level sets and strong cuts. The results here are generated by using the details of the deﬁnition of R. We also develop the ﬁrst subdirect product theorem and the metatheorem. The metatheorem in [15] was used to obtain results of other papers in a uniﬁed way. The second subdirect product theorem is applied to extend the modularity results of [3]. Deﬁne the function C : P(X) → C(X) by ∀Y ∈ P(X), C(Y )(x) = 1 if and only if x ∈ Y. Then C establishes a bijection of P(X) with C(X). For all Y ∈ P(X), C(Y ) is the characteristic function of Y. Every crisp subset is also a fuzzy subset. Recall that FP(X) has the partial order ⊆ deﬁned by ∀µ, ν ∈ FP(X), µ ⊆ ν if and only if µ(x) ≤ ν(x) ∀x ∈ X. Thus FP(X) is a complete lattice with C(X) a complete sublattice. P(X) is a complete lattice with respect to the partial order of containment and the function C : P(X) → John Mordeson, Kiran R. Bhutani, Azriel Rosenfeld: Fuzzy Group Theory, StudFuzz 182, 239– 266 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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C(X) is an isomorphism of complete lattices. The function S : C(X) → P(X) deﬁned by S(µ) = {x ∈ X | µ(x) = 1} is an isomorphism and is the inverse of C. By use of the functions C and S, it follows that every assertion about crisp subsets is also an assertion about (ordinary) subsets and conversely. We next deﬁne the representation function R. Recall that J = [0, 1). Deﬁnition 9.1.1. Let X be a nonempty set. Deﬁne the function R : FP(X) → C(X)J be deﬁned by ∀µ ∈ FP(X), ∀r ∈ J, and ∀x ∈ X, 0 if µ(x) ≤ r R(µ)(r)(x) = 1 otherwise. Proposition 9.1.2. The function R is injective. Proof. Suppose R(µ) = R(ν). Let x ∈ X. Since R(µ)(ν(x))(x) = R(ν)(ν(x))(x) = 0, it follows from the deﬁnition of R that µ(x) ≤ ν(x). Similarly, since R(ν)(µ(x))(x) = R(µ)(µ(x))(x) = 0, ν(x) ≤ µ(x). Thus µ(x) = ν(x) for all x ∈ X. Hence µ = ν.

Deﬁne the relation ≤ on C(X) as follows: ∀ G, H ∈ C(X) , G ≤ H if and only if ∀r ∈ J, G(r) ⊆ H(r). Since ⊆ is a partial order of C(X), ≤ is a partial order of C(X)J . It follows that the inequality, G(r) ⊆ H(r), holds if and only if ∀x ∈ X, G(r)(x) = 0 whenever H(r)(x) = 0. Equivalently, G(r) ⊆ H(r) holds if and only if H(r)(x) = 1 whenever G(r)(x) = 1. Further, observe that C(X)J has the least and the largest element K and T respectively given by: K(r)(x) = 0, T (r)(x) = 1 ∀r ∈ J and ∀x ∈ X. C(X)J is also closed under suprema and inﬁma and so C(X)J is indeed a bounded lattice. J

J

Proposition 9.1.3. Im(R) = {G : J → C(X) | ∀r ∈ J, G(r) = ∪{G(q) | q > r}}. Proof. Suppose that G = R(µ) for some µ ∈ F (X). Let r ∈ J and x ∈ X. Then G(r)(x) = R(µ)(r)(x) = 0 if and only if µ(x) ≤ r. However, µ(x) ≤ r if and only if µ(x) ≤ q for all q > r. Therefore, G(r)(x) = 0 if and only if G(q)(x) = 0 for all q > r. Thus G(r)(x) = ∨{G(q)(x) | q > r}. Since this equation holds ∀ x ∈ X, G(r) = ∪{G(q) | q > r} as required. Suppose that G : J → C(X) is such that ∀r ∈ J, G(r) = ∪{G(q) | q > r}. Deﬁne the function µ : X → I by ∀x ∈ X, µ(x) = ∧{p ∈ J | G(p)(x) = 0}. Then it follows that R(µ) = G since ∀r ∈ J and ∀x ∈ X, the following ﬁve assertions are equivalent: (1) (2) (3) (4) (5)

R(µ)(r)(x) = 0. µ(x) ≤ r. For all q > r, µ(x) ≤ q. For all q > r, G(q)(x) = 0. G(r)(x) = 0.

9.2 Representation of the Fuzzy Power Algebra

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It can be seen from Proposition 9.1.3 that ∀µ ∈ FP(X), R(µ) is an order reversing function from J into C(X), i.e., q ≥ r implies ∀x ∈ X, R(µ)(q)(x) ≤ R(µ)(r)(x). Proposition 9.1.4. R commutes with inﬁma of ﬁnite sets F of fuzzy subsets, i.e., R(∩µ∈F µ) = ∧{R(µ) | µ ∈ F } and with suprema of arbitrary sets S of fuzzy subsets, i.e., R(∪µ∈S µ) = ∨{R(µ) | µ ∈ S}. Proof. Let F be a ﬁnite collection of fuzzy subsets of X. That R(∩µ∈F µ) = ∧{R(µ) | µ ∈ F } can be seen from the fact that ∀r ∈ J and ∀x ∈ X, the following ﬁve statements are equivalent: (1) R(∩µ∈F µ)(r)(x) = 0. (2) (∩µ∈F µ)(x) ≤ r. (3) There exists µ ∈ F such that µ(x) ≤ r. (4) There exists µ ∈ F such that R(µ)(r)(x) = 0. (5) ∧{R(µ) | µ ∈ F }(r)(x) = 0. Let S be any collection of fuzzy subsets of X. That R(∪µ∈S µ) = ∨{R(µ) | µ ∈ S} can be seen from the fact that ∀r ∈ J and ∀x ∈ X, the following ﬁve statements are equivalent: (6) R(∪µ∈S µ)(r)(x) = 0. (7) (∪µ∈S µ)(x) ≤ r. (8) For all µ ∈ S, µ(x) ≤ r. (9) For all µ ∈ S, R(µ)(r)(x) = 0. (10) ∨{R(µ) | µ ∈ S}(r)(x) = 0. In the proof above, (2)⇒(3) fails when F is inﬁnite. It can be easily shown that R does not, in general, commute with inﬁma of a descending sequences of fuzzy subsets even when X is a singleton. As an example, let X = {a} be a set with one element. Deﬁne a countable family {µi }i∈N of fuzzy subsets of X as µi (a) = 1i ∀i ∈ N. Then ∩i∈N µi (a) = 0, but µi (a) = 0 ∀i ∈ N. Let I(X) be the image of the function R : FP(X) → C(X)J . By Proposition 9.1.4, I(X) is a sublattice and also a complete upper subsemilattice of C(X)J . By Propositions 9.1.2 and 9.1.4, R is an order isomorphism of FP(X) onto I(X). Note also that R0X = K and R1X = T where 0X (x) = 0 and 1X (x) = 1 ∀x ∈ X.

9.2 Representation of the Fuzzy Power Algebra In this section, we present the Subdirect Product Theorem. It plays an essential role in transferring crisp results to fuzzy ones. Deﬁnition 9.2.1. Let ∗ : X n → X be an n-ary operation on X, where n ∈ N. We deﬁne an n-ary operation ∗ on FP(X)n , ∗ : FP(X)n → FP(X) by the standard convolution method as follows: ∀µ1, ..., µn ∈ FP(X) and ∀x ∈ X, ∗(µ1 , . . . , µn )(x) = ∨{µ1 (x1 ) ∧ . . . ∧ µn (xn ) | x = ∗(x1 , . . . , xn )}.

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The operation ∗ on X also yields an n-ary operation ∗ on P(X) deﬁned by for all subsets X1 , . . . , Xn of X, ∗(X1 , . . . , Xn ) = {∗(x1 , . . . , xn ) | xi ∈ Xi , i = 1, 2, . . . , n}. Proposition 9.2.2. For every n-ary operation ∗ on X with n ∈ N, C(X) is closed with respect to the convolution extension of ∗ to FP(X). The bijection C : P(X) → C(X) commutes with the ∗ operation on P(X) and C(X). A convolution extension of ∗ from X to FP(X) induces an operation ∗ on C(X) which then provides a pointwise operation ∗ for C(X)J . Hence for G1 , . . . , Gn ∈ C(X)J and r ∈ J, ∗(G1 , . . . , Gn )(r) = ∗(G1 (r), . . . , Gn (r)). Proposition 9.2.3. For every n-ary operation ∗ on X, with n ∈ N, the representation function R : FP(X) → C(X)J commutes with the convolution extension of ∗, i.e.,∀µ1 , . . . , µn ∈ FP(X), R(∗(µ1 , . . . , µn )) = ∗(R(µ1 ), . . . , R(µn )). Proof. The desired result follows from the fact that ∀µ1 , . . . , µn ∈ FP(X), r ∈ J, and x ∈ X the following eight assertions are equivalent: (1) R(∗(µ1 , . . . , µn ))(r)(x) = 1. (2) ∗(µ1 , . . . , µn )(x) > r. (3) ∨{µ1 (x1 ) ∧ . . . ∧ µn (xn ) | x = ∗(x1 , . . . , xn )} > r. (4) There exists x1 , . . . , xn ∈ X for which x = ∗(x1 , . . . , xn ) and µ1 (x1 ) > r, . . . , µn (xn ) > r. (5) There exists x1 , . . . , xn ∈ X for which x = ∗(x1 , . . . , xn ) and R(µ1 )(r)(x1 ) = 1, . . . , R(µn )(r)(xn ) = 1. (6) ∨{R(µ1 )(r)(x1 ) ∧ . . . ∧ R(µn )(r)(xn ) | x = ∗(x1 , . . . , xn )} = 1. (7) ∗(R(µ1 )(r), . . . , R(µn )(r))(x) = 1. (8) ∗(R(µ1 ), . . . , R(µn ))(r)(x) = 1.

Now let X be an algebra of arbitrary structure. That is, X is a nonempty set provided with n-ary operations ∗1 , . . . , ∗k for various values of n ∈ N. Example 9.2.4. (1) X may be a semigroup having one binary operation, usually denoted by “·”. (2) X may be a group having one unary operation, usually denoted by −1 , and one binary operation, usually denoted by “·”. (3) X may be a ring having one unary operation, usually denoted by −, and two binary operations usually denoted by “+” and “·”. (4) X may be a lattice with two binary operations, “∨” and “∧”.

9.3 The Metatheorem

243

We have seen that the n-ary operations ∗1 , . . . , ∗k , n ≥ 1, extends to operations on P(X), C(X) and FP(X). Now P(X), C(X), and FP(X) have two additional binary operations, ∩ and ∪, as discussed previously. Thus P(X), C(X), and FP(X) become algebras having the corresponding operations ∩, ∪, ∗1 , . . . , ∗k . By Propositions 9.1.2, 9.1.4 and 9.2.3, it follows that R is an injective homomorphism of the algebra FP(X) into the product algebra C(X)J that establishes an algebraic and order-theoretic isomorphism of FP(X) with its image I(X). For all r ∈ J, the projection of FP(X) into the rth coordinate space of C(X)J is a surjection. In fact, ∀r ∈ J, R(µ)(r) maps the subset C(X) of FP(X) bijectively onto the rth coordinate space C(X) : Let µ ∈ C(X). Then ∀r ∈ J, x ∈ X, R(µ)(r)(x) = µ(x), i.e., R(µ)(r) = µ. Recall that an algebra A is said to be a subdirect product of a family of algebras {Ai | i ∈ I}, where I is an arbitrary index set, if A is isomorphic to a subalgebra of the product algebra Π{Ai | i ∈ I} with the property that its projection into each coordinate space Ai is a surjection. Now R deﬁnes an isomorphism of FP(X) onto I(X) and the projection of I(X) into each coordinate space of C(X)J is a surjection. Thus we have the following result. Theorem 9.2.5. (The Subdirect Product Theorem) Let X be an algebra having n-ary operations ∗1 , . . . , ∗k for various values of n ≥ 1. Then R : FP(X) → C(X)J is a representation of the (∩, ∪, ∗1 , . . . , ∗k )-algebra FP(X) as a subdirect product of copies of the (∩, ∪, ∗1 , . . . , ∗k )-algebra C(X).

9.3 The Metatheorem We review the recursive deﬁnition of an expression over a set V ar of variables, a set Op of operations, and the three auxiliary symbols left paren, right paren, and comma in order to eastablish our notation. Each operation symbol ∗ in Op has an integer n(∗) ≥ 1 as its arity. The following two sentences constitute the deﬁnition of an expression over V ar and Op : For all v ∈ V ar, v is an expression. For all ∗ ∈ Op and every list of expressions E1 , . . . , En(∗) , where n(∗) is the arity of ∗, ∗(E1 , . . . , En(∗) ) is an expression. Every expression is a ﬁnite string of symbols having a syntax rigidly speciﬁed by the recursive deﬁnition. Let E be an expression. Then the number of distinct symbols in V ar that occur in E is a positive integer and the number of distinct symbols in Op that occur in E is a nonnegative integer. Every symbol occurring in E may occur more than once. If no variable other than those in the set {v1 , . . . , vm } occurs in E, E may be denoted by writing E = E(v1 , . . . , vm ). The order in which the variable symbols are listed is immaterial and a variable symbol may occur in the list even though it does not occur in the expression E. The convenience of this notation is demonstrated in the next paragraph. Let X be a non-empty set provided with n-ary operations ∗1 , . . . , ∗k for various values of n ≥ 1. Then P(X), C(X), and FP(X) are algebras

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with operations ∩, ∪, ∗1 , . . . , ∗k . Let E = E(v1 , . . . , vm ) be an expression over the set of variables V ar = {v1 , . . . , vm } and the set of operations Op = {∩, ∪, ∗1 , . . . , ∗k }. Then for any m elements µ1 , . . . , µm ∈ FP(X), E(µ1 , . . . , µm ) is the element of FP(X) which results when, in E, each occurrence of each vi is replaced by µi and the result is evaluated in FP(X). We say that a class C of fuzzy subsets of X is closed under projections if ∀µ ∈ C, ∀r ∈ J, the crisp set R(µ)(r) is also in C. Theorem 9.3.1 (The Metatheorem). Let X be an algebra with n-ary operations ∗1 , . . . , ∗k for various values of n ≥ 1. Let FP(X) have the operations ∩, ∪, ∗1 , . . . , ∗k . Let D(v1 , . . . , vm ) and E(v1 , . . . , vm ) be expressions over the variable set {v1 , . . . , vm } and the operation set {∩, ∪, ∗1 , . . . , ∗k }. Let C1 , . . . , Cm be classes of fuzzy subsets of X that are closed under projections. The inequality D(µ1 , . . . , µm ) REL E(µ1 , . . . , µm ) holds for all fuzzy sets µi ∈ Ci , i = 1, 2, . . . , m if and only if it holds for all crisp sets µi ∈ Ci , i = 1, 2, . . . , m, where REL is any one of the three relations ⊆, =, or ⊇ . Proof. We consider only the case for which REL is ⊆ since the case for ⊇ follows by symmetry and then the case for = follows by combining ⊆ and ⊇ . Suppose the inequality D(µ1 , . . . , µm ) ⊆ E(µ1 , . . . , µm ) does not hold. Then ∃x ∈ X such that D(µ1 , . . . , µm )(x) > E(µ1 , . . . , µm )(x). Let r = E(µ1 , . . . , µm )(x). Then D(R(µ1 )(r), . . . , R(µm )(r))(x) = D(R(µ1 ), . . . , R(µm ))(r)(x) = R(D(µ1 , . . . , µm ))(r))(x) = 1. However, E(R(µ1 )(r), . . . , R(µm )(r))(x) = E(R(µ1 ), . . . , R(µm ))(r)(x) = R(E(µ1 , . . . , µm ))(r))(x) = 0. Let A be an (∧, ∨, ∗1 , . . . , ∗k )-algebra. For expressions D(v1 , . . . , vm ) and E(v1 , . . . , vm ) over V ar = {v1 , . . . , vm } and Op = {∧, ∨, ∗1 , . . . , ∗k }, we say that D(v1 , . . . , vm ) = E(v1 , . . . , vm ) is an (∧, ∨, ∗1 , . . . , ∗k )-identity. The algebra A satisﬁes the identity, D(v1 , . . . , vm ) = E(v1 , . . . , vm ) if ∀µ1 , . . . , µm ∈ A, D(µ1 , . . . , µm ) = E(µ1 , . . . , µm ). We obtain the following result when either the metatheorem or the subdirect product theorem is combined with the isomorphism of P(X) with C(X).

9.4 Uniﬁcations

245

Corollary 9.3.2. For every algebra X, with n-ary operations ∗1 , . . . , ∗k for various n ≥ 1, the three (∩, ∪, ∗1 , . . . , ∗k )-algebras P(X), C(X), and FP(X) all satisfy precisely the same (∩, ∪, ∗1 , . . . , ∗k )-identities. Let X be an algebra as in Corollary 9.3.2. Note that v1 ∨ v1 = 1, where the prime denotes the appropriate complementation, may be regarded as an identity that is satisﬁed by P(X) and C(X), but not by FP(X). However, v1 ∨ v1 = 1 is not an (∩, ∪, ∗1 , . . . , ∗k )-identity: Since 1 does not contain a variable, 1 is not an expression over V ar = {v1 } and Op = {∩, ∪, ∗1 , . . . , ∗k }. For this reason alone, v1 ∨v1 = 1 does not satisfy the deﬁnition of an (∩, ∪, ∗1 , . . . , ∗k )identity. However, even when X = P(Y ), X = C(Y ), or X = FP(Y ) for some set Y, it follows that the prime required in the identity is not the convolution extension of the complement appropriate to such an X. Corollary 9.3.2 provides two elementary known facts that are fundamental to the theory of fuzzy algebraic structures. One refers only to the information concerning ∩ and ∪ [34]: For every nonempty set X, the lattice of fuzzy subsets, FP(X), is distributive. Another refers to the operations ∗1 , . . . , ∗k and the identities they may satisfy. In some cases, identities satisﬁed by the (∗1 , . . . , ∗k )-algebra X are also satisﬁed by P(X). For example, this is true for the associativity identity and for the commutativity identity. Thus the metatheorem yields the following results: If (X, ∗) is a semigroup, then (FP(X), ∗) is a semigroup. If (X, ∗) is commutative, then so is (FP(X), ∗).

9.4 Uniﬁcations In this section, the classes Ci in the metatheorem are required to satisfy restrictions that are closely related to the structure of the (∗1 , . . . , ∗k )-algebra X. That these conditions are easily dealt with is a consequence of the following property of R which follows immediately from its deﬁnition. Proposition 9.4.1. For all x, y ∈ X and ∀µ ∈ FP(X), µ(x) REL µ(y) holds if and only if ∀r ∈ J, R(µ)(r)(x) REL R(µ)(r)(y) holds, where REL is any one of the three relations ≤, =, or ≥ . Several fundamental results in the literature can be obtained by noting that they are consequences of Proposition 9.4.1. Generic patterns for deﬁning fuzzy versions of crisp concepts will allow many results to be treated simultaneously. For each class C of crisp subsets of X, a fuzzy subset µ ∈ FP(X) is called a fuzzy C subset if and only if ∀r ∈ J, R(µ)(r) ∈ C. This generic definition is consistent with many deﬁnitions from semigroup theory as pointed out in [15]. We do not discuss them here since they do not fall within the framework of group theory. The convenience of this general concept of a fuzzy C class is illustrated by deriving generalizations of results in a uniﬁed manner by reference to the following principle which is immediate from the deﬁnition of a fuzzy C class: For classes C and D of crisp subsets, the class of fuzzy C

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subsets is contained in the class of fuzzy D subsets if and only if C is contained in D. From this principle and the observations made above, numerous propositions hold [15]. We now give an application to lattice modularity. Let (X,−1 , ∗) be a group. Recall that N F(X) denotes the set of all normal fuzzy subgroups in FP(X). With respect to the partial order, ⊆, of FP(X), N F (X) is a lattice for which the INF coincides with the inf of the lattice FP(X), but for which the SUP does not. By normality, this SUP is very well behaved: SU P (µ, ν) = µ ∗ ν ∀µ, ν ∈ N F(X). In the present exposition, the fact that the SUP of {µ, ν} is µ ∗ ν, is a consequence of Proposition 9.4.1 and the deﬁnition of a normal fuzzy subgroup: Since µ and ν are normal fuzzy subgroups, R(µ)(r) and R(ν)(r) are (crisp) normal subgroups ∀r ∈ J. Since R commutes with ∗, R(µ ∗ ν)(r) = R(µ)(r) ∗ R(ν)(r) holds ∀r ∈ J and is the SUP of its two displayed crisp factors. Since R(µ∗ν)(r) is normal ∀r ∈ J, µ∗ν is normal. Since R(µ ∗ ν)(r) = SU P {R(µ)(r), R(ν)(r)} ∀r, µ ∗ ν = SU P {µ, ν} if and only if µ(e) = ν(e). It is shown in Proposition 9.1.4 that R commutes with SUP. Corollary 9.4.2. Let (X,−1 , ∗) be a group. Let N F(X) be the lattice of normal fuzzy subgroups in FP(X). Let LR be the restriction of R : FP(X) → C(X)J to N F (X). Let LI(X) denote the image of LR and let LC consist of the crisp subsets in N F(X). Then LR is an isomorphism of the lattice N F (X) onto LI(X) that represents N F(X) as a subdirect product in the lattice LC J . Proof. This result follows from the subdirect product theorem since from that theorem we know that R is an injection that commutes with ∩ and ∗. Since N F (X) is closed under ∩ and ∗, LR is also an injection that commutes with ∩ and ∗. Since ∗ provides the SU P operation of N F(X), the desired result follows. From Corollary 9.4.2, we obtain the following result. Corollary 9.4.3. For each group (X,−1 , ∗) the lattice of normal fuzzy subgroups and the lattice of crisp normal subgroups satisfy precisely the same (∩ = IN F, ∗ = SU P )-identities. A lattice is modular if and only if satisﬁes the identity: v1 ∨ (v2 ∧ (v1 ∨ v3 )) = (v1 ∨ v2 ) ∧ (v1 ∨ v3 ) Since the lattice of normal subgroups of a group is known to be modular, we have from Corollary 9.4.3 that for every group (X,−1 , ∗) the lattice of normal fuzzy subgroups is modular. Let LF consist of the fuzzy normal subgroups µ ∈ FP(X) for which the set {µ(x) | x ∈ X} is ﬁnite. Let the identity element of the group X be e and let t ∈ [0, 1]. Let LF t = {µ ∈ LF | µ(e) = t}. Both ∧ and ∗ preserve the

9.5 Lattices of Fuzzy Congruences

247

property of having a ﬁnite image set and the property of assuming a given value at e. Consequently, LF and all of the LF t are sublattices of the lattice of normal fuzzy subgroups in FP(X). This observation gives the following result [[6], Theorem 3.10]: For each group (X,−1 , ∗), the lattice LF is modular and as is LF t, ∀t ∈ [0, 1]. When the concept of a fuzzy characteristic subgroup and a fuzzy fully invariant subgroup are deﬁned generically, we have immediately that for each group (X,−1 , ∗), the fuzzy characteristic subgroups (respectively, the fuzzy fully invariant subgroups) form a sublattice of the lattice of normal fuzzy subgroups and so must also be modular. The metatheorem and the subdirect product theorems given here apply to the study of other fuzzy algebraic structures.

9.5 Lattices of Fuzzy Congruences Most of the results in this and the next section are from [10]. Let L(G) denote F(G) and Ln (G) denote N F(G) in the remainder of the chapter, where G is a group. In this section, we discuss certain properties of fuzzy congruences and fuzzy normal subgroups of a group in relation to strong level subsets. We provide generating techniques for fuzzy congruences and fuzzy subgroups of a group. With these techniques, we prove that the lattices of fuzzy congruences and fuzzy normal subgroups of a group are isomorphic. This is a generalization of a result that the lattices of t-fuzzy congruences and fuzzy normal subgroups with tip “t” are isomorphic. We establish this isomorphism by following the metatheorem approach presented earlier in the chapter. We use a function that represents the algebra of fuzzy subsets of a set as a subdirect product of copies of the algebra of crisp subsets. In [2], the use of the notion of level subsets was shifted to that of strong level subsets. Strong level subsets characterizations are used more eﬀectively to obtain results in fuzzy group theory and other fuzzy algebraic structures. This approach provides a method for fuzziﬁcation of results and ideas from classical algebra to fuzzy algebraic structures. In the present section, we follow this approach to investigate the properties of fuzzy congruences and fuzzy normal subgroups of a group. Fuzzy equivalence relations and fuzzy congruences have been studied in [26, 27]. In [31], the foundation of lattice theoretic studies in the area of fuzzy group theory was laid. It was considered further in the papers [3, 6, 8]. Lattice theoretic results of fuzzy subgroups in a more general setting were presented in [23, 24]. The purpose of [6] was to demonstrate that fuzzy algebraic structures are important when viewed from a lattice theoretic point of view. For this reason, the sublattices Lt , Lf , Lf t , Lf nt and Lnt of the lattice L = L(G) of all fuzzy subgroups of a given group G were examined. Also, in [8] various types of sublattices of the lattice E of all fuzzy equivalence relations on a group were constructed. In particular, the sublattices Ct, Cf , Cf t and Cst of the lattice C

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of all fuzzy congruence relations were examined. Furthermore, it was shown that the lattice Ct of all t-fuzzy congruences and the lattice Lnt of all fuzzy normal subgroups with tip “t” are isomorphic for all t ∈ (0, 1]. Moreover, in [3] it was established that the lattice Ln of all fuzzy normal subgroups is a sublattice of the lattice L of all fuzzy subgroups of a given group. As in classical group theory, an isomorphism between lattices C and Ln of a given group is presented. The isomorphism between these lattices is established by using the approach of strong level subsets. There is a close link between the approach used here and that of the metatheorem approach given in Sections 9.1-9.4. Let G be a group. For µ ∈ FP(G) and α ∈ FP(G × G)), we let ∨µ denote ∨{µ(x) | x ∈ G} and ∨α denote ∨{α(x, y) | (x, y) ∈ G × G)}. Let µ ∈ FP(G). For a fuzzy subgroup µ, we call µ(e) the tip of the fuzzy subgroup µ. Let Lt (G) denote the set of all fuzzy subgroups of G with tip “t”. Let ν ∈ L(G). Recall that ν is called normal if and only if ν(xy) = ν(yx) ∀x, y ∈ G. We let Ln (G) denote the set of all normal fuzzy subgroups of G and Lnt (G) the set of all normal fuzzy subgroups in Lt (G). Let α ∈ FP(G × G). For a ﬁxed t ∈ [0, 1], α is called a t-fuzzy relation if ∨α = t. A t-fuzzy relation α is called t-reﬂexive if α(x, x) = t, ∀x ∈ G. A t-reﬂexive relation α is called a t-fuzzy equivalence relation if α is symmetric, that is, α(x, y) = α(y, x)∀x, y ∈ G and α is sup-min transitive, that is, α(x, y) ≥ ∨{α(x, z) ∧ α(z, y) | z ∈ G}∀x, y ∈ G. Let Et (G) denote the set of all t-fuzzy equivalence relations on G and let E(G) denote the set of all fuzzy equivalence relations on G, i.e., E(G) = ∪t∈[0,1] Et (G). Let β ∈ Et (G). Then β is called a t-fuzzy congruence relation if β(ac, bd) ≥ β(a, b) ∧ β(c, d), ∀a, b, c, d ∈ G. Let Ct (G) denote the set of all t-fuzzy congruence relations in Et (G) and C(G) denote the set of all fuzzy congruence relations in E(G), i.e., C(G) = ∪t∈[0,1] Ct (G). For µ ∈ FP(G), the fuzzy subgroup generated by µ is deﬁned to be the least member of L(G) containing µ and is denoted by µ. We use the same notation for an ordinary subgroup generated by the level subset (strong level subset) of µ. Similarly for α ∈ FP(G × G) a fuzzy equivalence relation generated by α is denoted by [α] and a fuzzy congruence relation generated by α is denoted by &α'. Also, the same notations are used for ordinary equivalence relations and congruence relations generated by level subsets (strong level subsets) of α. Let µ ∈ FP(G). We assume in the following that ∨µ > 0. Proposition 9.5.1. Let µ ∈ FP(G) and t0 = ∨µ. Then the following conditions are equivalent: (1) µ ∈ L(G). (2) µ> t is a subgroup of G ∀t ∈ [0, t0 ). (3) Every nonempty strong level subset of µ is a subgroup of G.

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249

> Proof. (1) ⇒ (2) : Let t ∈ [0, t0 ). Then µ> t = ∅. Let x, y ∈ µt . Then µ(x) > t −1 and µ(y) > t. Since µ is a fuzzy subgroup of G, µ(xy ) ≥ µ(x) ∧ µ(y) > t. > Thus xy −1 ∈ µ> t . Hence µt is a subgroup of G. > (2) ⇒ (3) : Let µt be a nonempty strong level subset of G, where t ∈ [0, 1]. > Then there exists x ∈ µ> t and for any such x, t < µ(x) ≤ t0 . Hence by (2), µt is a subgroup of G. (3) ⇒ (1) : Let x, y ∈ G and µ(x) = t1 and µ(y) = t2 . If either t1 = 0 or t2 = 0, then µ(xy −1 ) ≥ 0 = µ(x) ∧ µ(y). Suppose both t1 > 0 and t2 > 0. Let > −1 ∈ µ> 0 ≤ t < t1 ∧ t2 . Then x, y ∈ µ> t . Since µt is a subgroup of G, xy t and −1 −1 so µ(xy ) > t. Thus µ(xy ) ≥ t1 ∧ t2 = µ(x) ∧ µ(y). Hence µ is a fuzzy subgroup of G.

Proposition 9.5.2. Let µ ∈ L(G). Then the following conditions are equivalent: (1) µ ∈ Ln (G). (2) µ> t is a normal subgroup of G, ∀t ∈ [0, t0 ), where t0 = µ(e). (3) Every nonempty strong level subgroup is a normal subgroup of G. Proof. (1) ⇒ (2) : Let x ∈ G and y ∈ µ> t , where t ∈ [0, t0 ). Since µ is normal, > µ(xyx−1 ) = µ(y) > t. Thus µxyx−1 ∈ µ> t and so µt is normal. > (2) ⇒ (3) : Let µt be a nonempty strong level subset of G, where t ∈ [0, 1]. > Then there exists x ∈ µ> t and for any such x, t < µ(x) ≤ t0 . Hence by (2), µt is a normal subgroup of G. (3) ⇒ (1) : Let x, y ∈ G and µ(y) = t2 . Let 0 ≤ t < t2 . Then y ∈ µ> t . −1 −1 ∈ µ> ) > t. Hence Since µ> t is a normal subgroup of G, xyx t . Thus µ(xyx µ(xyx−1 ) ≥ µ(y). Thus µ is normal. Theorem 9.5.3. Let α ∈ FP(G × G). Then the following conditions are equivalent: (1) α ∈ C(G). (2) αt> is a congruence relation on G ∀t ∈ [0, t0 ), where t0 = ∨α. (3) Every nonempty strong level subset is a congruence relation on G. Proof. (1) ⇒ (2) : Let t ∈ [0, t0 ). For all x ∈ G, α(x, x) = t0 > t. Thus (x, x) ∈ αt> . Suppose (x, y) ∈ αt> . Then α(y, x) = α(x, y) > t and so (y, x) ∈ αt> . Suppose (x, z), (z, y) ∈ αt> . Then α(x, y) ≥ ∨{α(x, z ) ∧ α(z , y) | z ∈ G} ≥ α(x, z) ∧ α(z, y) > t. Thus (x, y) ∈ αt> . Hence αt> is an equivalence relation. Let (a, b), (c, d) ∈ αt> . Then α(ac, bd) ≥ α(a, b) ∧ α(c, d)˙ > t. Thus (ac, bd) ∈ αt> . Therefore, αt> is a congruence relation on G. (2) ⇒ (3) : Suppose αt> = ∅, where t ∈ [0, 1]. Then there exists (x, y) ∈ αt> and for all such (x, y), t < α(x, y) ≤ t0 . Hence by (2), αt> is an ordinary congruence relation on G. (3) ⇒ (1) : For all t ∈ [0, 1] such that αt> = ∅, (x, x) ∈ αt> for all x ∈ G. Thus it follows that α(x, x) = t0 for all x ∈ G, where t0 = ∨α. Hence αt> = ∅ for all t such that t ∈ [0, t0 ). Let t ∈ [0, t0 ).Then αt> is a congruence relation on G. Let x, y ∈ G. Since (x, y) ∈ αt> if and only if (y, x) ∈ αt>

250

9 Lattices of Fuzzy Subgroups

for all t ∈ [0, t0 ), it follows that α(x, y) = α(y, x). Let (x, z), (z, y) ∈ G. Then α(x, z) > t and α(z, y) > t ⇒ (x, z), (z, y) ∈ αt> ⇒ (x, y) ∈ αt> ⇒ α(x, y) > t ⇒ α(x, y) ≥ ∨{α(x, z)∧α(z, y) | z ∈ G}. Hence α ∈ E(G). Suppose (a, b), (c, d) ∈ G × G. Then α(a, b)˙ > t, α(c, d)˙ > t ⇒ (a, b), (c, d) ∈ αt> ⇒ (ac, bd) ∈ αt> ⇒ α(ac, bd) > t. Therefore, α(ac, bd) ≥ α(a, b) ∧ α(c, d). Thus α ∈ C(G). Theorem 9.5.4. Let α ∈ FP(G × G). Deﬁne a fuzzy relation α∗ on G by α∗ (x, y) = ∨{r | (x, y) ∈ [αr ], r < ∨α}, ∀(x, y) ∈ G × G. ∗ Then α = [α]. Proof. We ﬁrst show that α∗ is a fuzzy equivalence relation. Let x ∈ G. Then α∗ (x, x) = t0 since (x, x) ∈ [αr ] ∀r < t0 , where t0 = ∨α. F or all x, y ∈ G, α∗ (x, y) = α∗ (y, x) since (x, y) ∈ [αr ] ⇔ (y, x) ∈ [αr ]. Let x, y, z ∈ G and let t1 = α∗ (x, z) and t2 = α∗ (z, y). Then (x, z) ∈ [αr1 ] ∀r1 < t1 and (z, y) ∈ [αr2 ] ∀ r2 < t2 . Suppose t1 ≥ t2 . (A similar argument can be used if t1 ≤ t2 .) Hence (x, z), (z, y) ∈ [αr2 ] ∀ r2 < t2 . Thus (x, y) ∈ [αr2 ] ∀ r2 , t2 . Thus α∗ (x, y) ≥ t2 = α∗ (x, z) ∧ α∗ (z, y). Thus it follows that α∗ is a fuzzy equivalence relation. Let x, y ∈ G and α(x, y) = r. Then (x, y) ∈ αr ⊆ [αr ]. Thus α∗ (x, y) ≥ r. Hence α ⊆ α∗ . Thus [α] ⊆ α∗ . We show that [α] = α∗ by showing that very fuzzy equivalence relation which contains α contains α∗ . Let θ be a fuzzy equivalence relation on G such that θ ⊇ α. Then θr ⊇ αr and so θr ⊇ [αr ] ∀ r ∈ [0, t0 ). Let α∗ (x, y) = r. Then (x, y) ∈ [αr− ] ⊆ θr− ∀ such that 0 < < r. Hence θ(x, y) ≥ r − ∀ such that 0 < < r. Thus θ(x, y) ≥ r. Hence α∗ ⊆ θ. Therefore, α∗ = [α]. Theorem 9.5.5. Let α ∈ FP(G × G). Deﬁne fuzzy relations α∗ and α∗∗ on G by ∀(x, y) ∈ G × G, α∗ (x, y) = ∨{r | (x, y) ∈ &αr ', r < ∨α}, α∗∗ (x, y) = ∨{r | (x, y) ∈ &αr> ', r < ∨α}. ∗ Then α = α∗∗ = &α'. Proof. First, we show that α∗∗ ∈ C(G). Let t0 = ∨α. Then by the deﬁnition ∗∗ is empty. of α∗∗ , for all t ∈ [t0 , 1), the strong level subset (α∗∗ )> t of α We show that > (α∗∗ )> t = &αt ' ∀t ∈ [0, t0 ). ∗∗ > ∗∗ Let (x, y) ∈ (α )t . Then α (x, y) > t. Thus ∨{r | (x, y) ∈ &αr> ', r < ∨α} > t. Hence ∃s ∈ [0, t0 ) such that s > t and (x, y) ∈ &αs> '. However, > > &αs> ' ⊆ &αt> ' and so (x, y) ∈ &αt> '. Thus (α∗∗ )> t ⊆ &αt '. Let (x, y) ∈ &αt '. > > Now an ordinary congruence relation &αt ' generated by the subset αt in a semigroup (or group) can be written as &αt> ' = [(αt> )c ] = [αt> ]c ,

(9.5.1)

where (αt> )c and [αt> ]c are the smallest compatible relations containing αt> and [αt> ], respectively. Therefore, (x, y) ∈ [αt> ]c . Hence ∃(x1, y1 ) ∈ G × G and a, b ∈ G such that

9.5 Lattices of Fuzzy Congruences

(x1 , y1 ) ∈ [αt> ] and (ax1 b, ay1 b) = (x, y)

251

(9.5.2)

Thus ∃x1 = z0 , z1 , z2 , ..., zn−1 , zn = y1 in G such that α(zi−1 , zi ) > t, ∀i = 1, 2, ..., n. Let t1 = ∧{α(zi−1 , zi ) | 1 ≤ i ≤ n}. Then t1 > t. Let t such be that t1 > t > t. Then (zi−1, zi ) ∈ αt> , ∀i = 1, 2, ..., n. Thus (x1, y1 ) ∈ [αt> ] ⊆ [αt> ]c . Therefore, by the above Expressions 9.5.1 and 9.5.2 it follows that (x, y) ∈ &αt> '. Hence ∨{r | (x, y) ∈ &αr> ', r < ∨α} ≥ t > t. Thus by deﬁnition of α∗∗ , we have (x, y) ∈ (α∗∗ )> t . Hence > (α∗∗ )> t = &αt ', ∀t ∈ [0, t0 )

(9.5.3)

By Theorem 9.5.3(1), α∗∗ ∈ C(G) since each nonempty strong level subset of α∗∗ is an ordinary congruence relation generated by the relation αt> . Now ∀t ∈ [0, t0 ), αt> ⊆ &αt> ' and ∀t ∈ [t0, 1), αt> is empty. However, > &αt ' = {(x, x) | x ∈ G}. Hence αt> ⊆ &αt> ', ∀t ∈ [0, 1). Therefore, by Expression 9.5.3 above, we also have α ⊆ α∗∗ . Let θ ∈ C(G) be such that α ⊆ θ. Then αt> ⊆ θt> ∀t ∈ [0, 1). Thus > > &αt ' ⊆ θt> . Therefore, by Expression 9.5.3, we have (α∗∗ )> t ⊆ θt ∀t ∈ [0, 1). ∗∗ ∗∗ Hence α ⊆ θ and so α = &α'. Finally, we show that the fuzzy subsets α∗ and α∗∗ are identical. By their deﬁnitions, it is clear that α∗∗ ⊆ α∗ . To show the reverse inclusion, let t ∈ ∗ > [0, 1). Now if t ∈ [t0 , 1), then (α∗∗ )> t and (α )t are empty. On the other ∗ > hand, for t ∈ [0, t0 ), we let (x, y) ∈ (α )t . Then by deﬁnition of α∗ , ∃k > t such that (x, y) ∈ &αk '. However, &αk ' ⊆ &αt> ' and so (x, y) ∈ &αt> '. Hence > > ∗ > ∗∗ > (α∗ )> t ⊆ &αt '. Therefore, by Expression 9.5.3, (α )t ⊆ &αt ' = (α )t , ∗∗ > ∗ ∗∗ ⊆ (α ) , ∀t ∈ [0, 1). Thus α ⊆ α . ∀t ∈ [0, t0 ). Hence (α∗ )> t t The technique used in Theorem 9.5.4 and Theorem 9.5.5 can also be used to obtain a similar construction of a fuzzy subgroup generated by an arbitrary fuzzy subset as can be seen in the following result. Theorem 9.5.6. Let µ ∈ FP(G). Deﬁne fuzzy subsets µ∗ and µ∗∗ of G by ∀x ∈ G, (1) µ∗ (x) = ∨{r | x ∈ µr , r < ∨µ}, (2) µ∗∗ (x) = ∨{r | x ∈ µ> r , r < ∨µ}. Then µ∗ = µ∗∗ = µ. Proof. We have that (1) holds since (1) is a restatement of Proposition 5.3.4. > ∗∗ For all r < ∨µ, µ> ⊆ µ∗ = µ. That r ⊆ µr and so µr ⊆ µr . Thus µ µ ⊆ µ∗∗ and µ∗∗ is a fuzzy subgroup of G follows as in the proof of Proposition 5.3.4. Hence (2) holds. Lemma 9.5.7. Let α ∈ FP(G × G) and µ ∈ FP(G). Then the following assertions hold. (1) &α'(x, x) = ∨{α(y, z) | (y, z) ∈ G × G} ∀x ∈ G. (2) µ(e) = ∨{µ(x) | x ∈ G}.

252

9 Lattices of Fuzzy Subgroups

Lemma 9.5.8. Let α ∈ FP(G × G) and µ ∈ FP(G). Then > (1) &α'> t = &αt ', ∀t ∈ [0, t0 ), where t0 = α(x, x), > > (2) µt = µt , ∀t ∈ [0, s0 ), where s0 = (µ)(e). Lemma 9.5.9. Let α, β ∈ C(G) and µ, ν ∈ L(G). Then > > (1) &α ∪ β'> t = &αt ∪ βt ' ∀t ∈ [0, t0 ), where t0 = α(x, x) ∨ β(x, x), > > > (2) µ ∪ νt = µt ∪ νt ∀t ∈ [0, s0 ), where s0 = µ(e) ∨ ν(e). Theorem 9.5.10. Let µ and ν be normal fuzzy subgroups of G. Then the fuzzy subgroup µ ∪ ν is normal. Proof. By Lemma 9.5.7, µ ∪ ν(e) = ∨(µ ∪ ν(x) | x ∈ G} = ∨{µ(x) ∨ ν(x) | x ∈ G} = (∨{µ(x) | x ∈ G}) ∨ (∨{ν(x) | x ∈ G}) = µ(e) ∨ ν(e). By > > Lemma 9.5.9, µ ∪ ν> t = µt ∪ νt ∀t ∈ [0, s0 ), where s0 = µ(e) ∨ ν(e). Let p = µ(e) ∧ ν(e) and q = µ(e) ∨ ν(e). Suppose t ∈ [0, p). Then by Lemma > > > 9.5.9,µ ∪ ν> t = µt ∪ νt , ∀t ∈ [0, p). Since µ and ν are normal, µt and > νt are normal subgroups of G ∀t ∈ [0, p) by Proposition 9.5.2. Hence the > product µ> t νt is a normal subgroup of G ∀t ∈ [0, p). Suppose t ∈ [p, q). Then > > > > µ ∪ νt = µ> t or νt (= µt or νt ), respectively, ∀t ∈ [0, s0 ). Combining > the two cases, we have µ ∪ νt is a normal subgroup of G∀t ∈ [0, s0 ). Hence by Proposition 9.5.2, µ ∪ ν is a normal fuzzy subgroup of G. In [2, 3, 6, 8] various sublattices of the lattices of fuzzy subgroups and fuzzy equivalence relations for a given group were discussed. We recall the following result. Theorem 9.5.11. The set Ln (G) forms a complete sublattice of the lattice L(G). Proof. Let µ, ν ∈ Ln (G). Then it follows that the join of µ and ν, µ ∨ ν, is µ ∪ ν. By the previous theorem, µ ∨ ν ∈ Ln (G). The meet of µ and ν, µ ∧ ν, is µ ∩ ν. Thus µ ∧ ν ∈ Ln (G). In fact, the intersection of any collection of normal fuzzy subgroups of G is a normal fuzzy subgroup of G. The following result can also be easily veriﬁed. Theorem 9.5.12. The set C(G) forms a complete sublattice of the lattice E(G). It is known that there exists a one-to-one correspondence between the set of all congruence relations on a group and the set of all normal subgroups of that group. For a congruence relation R on a group G, a normal subgroup N (R) can be deﬁned as follows: N (R) = {x ∈ G|(x, e) ∈ R},

(9.5.4)

where e is the identity of G. For a normal subgroup H of G, a congruence relation C(H) can deﬁned as follows:

9.5 Lattices of Fuzzy Congruences

C(H) = {(x, y) ∈ G × G|xy −1 ∈ H}.

253

(9.5.5)

It is well known that the function φ from the set of all congruence relations on G onto the set of all normal subgroups of G given by ϕ(R) = N (R) not only establishes a one-to-one correspondence, but also provides an isomorphism between the lattices of congruence relations on G and its normal subgroups. The inverse of this isomorphism is given by φ−1 (H) = C(H). For all congruence relations R1, R2 on G, N (&R1 ∪ R2 ') = (N (R1 ) ∪ N (R2 ))

(9.5.6)

N (R1 ∩ R2 ) = N (R1 ) ∩ N (R2 ).

(9.5.7)

and

For a fuzzy congruence relation β on a group G, deﬁne a fuzzy subset N (β) of G as follows: N (β)(x) = β(x, e), ∀x ∈ G.

(9.5.8)

Also, for a normal fuzzy subgroup θ of G, deﬁne a fuzzy relation C(θ) in G as follows: C(θ)(x, y) = θ(xy −1 ), ∀(x, y) ∈ G × G.

(9.5.9)

In order to show that the fuzzy sets N (β) and C(θ) play the role of subsets given in Expressions 9.5.4 and 9.5.5 respectively, we prove the following result. Lemma 9.5.13. Let β ∈ C(G) and θ ∈ Ln (G) and let N (β) and C(θ) be as deﬁned in the above Expressions 9.5.8 and 9.5.9, respectively. Then the following assertions hold. > (1) (N (β))> t = N (βt ), ∀t ∈ [0, 1). > > (2) (C(θ))t = C(θt ), ∀t ∈ [0, 1). Proof. (1) Let t ∈ [0, 1). Then for x ∈ G, we have that x ∈ (N (β))> t ⇔ N (β)(x) > t ⇔ β(x, e) > t (by deﬁnition) ⇔ (x, e) ∈ βt> ⇔ x ∈ N (βt> ) (by Expression 9.5.4). Thus (1) holds. (2) Let t ∈ [0, 1). Then for (x, y) ∈ G×G, we have that (x, y) ∈ (C(θ))> t ⇔ C(θ)(x, y) > t ⇔ θ(xy −1 ) > t (by deﬁnition) ⇔ xy −1 ∈ θt> ⇔ (x, y) ∈ C(θt> ) (by Expression 9.5.5). Thus (2) holds. Theorem 9.5.14. [8, 19, 22, 32] Let β ∈ C(G). Then N (β) ∈ Ln (G). Proof. Suppose β ∈ C(G). Then β(e, e) = ∨β. Let t0 = ∨β. Then N (β)(e) = t0 . Thus β ∈ C(G) ⇔ βt> is a congruence relation on G for all t ∈ [0, t0 ) (by Theorem 9.5.3) ⇒ N (βt> ) is a normal subgroup of G for all t ∈ [0, t0 ) ⇔ (N (β))> t is a normal subgroup of G for all t ∈ [0, t0 ) ⇔ β is a normal fuzzy subgroup of G (by Proposition 9.5.2).

254

9 Lattices of Fuzzy Subgroups

Theorem 9.5.15. [8, 19, 22, 32] Let θ ∈ Ln (G). Then C(θ) ∈ C(G). Proof. Suppose θ ∈ Ln (G). Then for all x ∈ G, C(θ)(x, x) = θ(xx−1 ) = θ(e). Let t0 = θ(e). Then ∨C(θ) = t0 . Thus θ ∈ Ln (G) ⇔ θt> is a normal subgroup of G for all t ∈ [0, t0 ) (by Proposition 9.5.2) ⇒ C(θt> ) is a congruence relation on G for all t ∈ [0, t0 ) ⇔ (C(θ))> t is a congruence relation on G for all t ∈ [0, t0 ) (since C(θt> ) = {(x, y)|xy −1 ∈ θt> } = {(x, y)|θ(x, y −1 ) > t} = {(x, y)|C(θ)(x, y) > t} = C(θ)> t ) ⇔ C(θ) is a fuzzy congruence relation on G (by Theorem 9.5.3). Theorem 9.5.16. Let β ∈ C(G) and θ ∈ Ln (G). Let N (β) and C(θ) be deﬁned as in Expressions 9.5.8 and 9.5.9, respectively. Then the following assertions hold: (1) C(N (β)) = β. (2) N (C(θ)) = θ. Proof. (1) Let x, y ∈ G. Then C(N (β))(x, y) = N (β)(xy −1 ) = β(xy −1 , yy −1 ) ≥ β(x, y) ∧ β(y −1 , y −1 ) = β(x, y). Now β(x, y) = β(xy −1 y, ey) ≥ β(xy −1 , e) ∧ β(y, y) = β(xy −1 , e) = N (β)(xy −1 ) = C(N (β))(x, y). Hence C(N (β))(x, y) = β(x, y). (2) Let x ∈ G. Then N (C(θ))(x) = C(θ)(x, e) = θ(xe−1 ) = θ(x).

Theorem 9.5.17. The lattices of fuzzy congruences and fuzzy normal subgroups of group G are isomorphic. That is, C(G) ∼ = Ln (G). Proof. Deﬁne the function f : C(G) → Ln (G) by f (α) = N (α), ∀α ∈ C(G). Let θ ∈ Ln (G). Then by Theorem 9.5.15, C(θ) ∈ C(G) and so by Theorem 9.5.16 (2), N (C(θ)) = θ. Thus f is onto. Let β, γ ∈ C(G) be such that f (β) = f (γ). Then N (β) = N (γ) . It follows easily that ∨β = ∨N (β) and ∨γ = ∨N (γ). Hence ∨γ = ∨β. Thus by Lemma 9.5.13 (1), N (βt> ) = N (γt> ) ∀t ∈ [0, 1). Therefore, C(N (βt> )) = C(N (γt> )) and hence βt> = γt> , ∀t ∈ [0, 1). Consequently, β = γ. Thus f is one-to-one. Finally, we show that the function f preserves ∨ (join) and ∧ (meet). Let β, γ ∈ C(G). We prove that N (&β ∪ γ') = N (β) ∪ N (γ) and N (β ∩ γ) = N (β) ∩ N (γ). Before doing so, we note that ∨&β ∪γ' = (∨β)∨(∨γ) and so ∨N (&β ∪γ') = (∨β)∨(∨γ). Also, ∨&N (β)∪N (γ)' = (∨N (β))∨(∨N (γ)} = (∨β)∨(∨γ). Since N (&β ∪ γ') and (N (β) ∪ N (γ)) have the same supremum, say t0, it follows

9.6 The Metatheorem Approach

255

> that N (&β ∪ γ')> t = φ = (N (β) ∪ N (γ))t ∀t ∈ [t0 , 1). On the other hand, ∀t ∈ [0, t0 ), we have that > by Lemma 9.5.8 N (&β ∪ γ')> t = N (&β ∪ γ't ) > by Lemma 9.5.9 = N (&βt ∪ γt> ') using Expression 9.5.6 = (N (βt> ) ∪ N (γt> )) > by Lemma 9.5.13 (1) = ((N (β))> t ∪ (N (γ))t ) > by Lemma 9.5.9 (2). = (N (β) ∪ N (γ))t Therefore, N (&β ∪ γ') = (N (β) ∪ N (γ)). By a similar argument, it follows that > (N (β ∩ γ))> t = (N (β) ∩ N (γ))t , ∀t ∈ [0, 1). Thus N (β ∩ γ) = N (β) ∩ N (γ).

9.6 The Metatheorem Approach We now consider the metatheorem approach. In the following, we use the subdirect product theorem developed in Section 9.4 to establish our main result. The technique used here and also applied in [15] demonstrates the ease in obtaining the fuzzy versions of the results from their crisp counterparts. In fact, our main Isomorphism Theorem is established here without using the join formula of the lattice of fuzzy normal subgroups and that of fuzzy congruences in a group. We have the following corollary of Proposition 9.2.3. Let X be a semigroup with operation ∗. Then the systems (P(X), ∗) and (C(X), ∗) are isomorphic semigroups. Let (X,−1 , ∗) be a group. The set Ln of all normal fuzzy subgroups of X is a lattice under the ordering ⊆ of FP(X), the inf in Ln coincides with the inf of the lattice FP(X), whereas the SUP is deﬁned as in Theorem 9.5.6. Moreover, we have µ ∨ ν = (µ ∪ ν) = µ ∗ ν , where µ and ν are the tip extended pair of fuzzy subgroups obtained by µ and ν respectively. That is, µ (e) = ν (e) = µ(e) ∨ ν(e) and µ (x) = µ(x) and ν (x) = ν(x) of all x ∈ G, x = e. Let µ, ν ∈ Ln . Then µ , ν ∈ Ln . Hence it is immediate that µ ∗ ν is a normal fuzzy subgroup. Next, let R|Ln be the restriction of the function R : FP(X) → C(X)J to Ln and let LI(X) be the image of R|Ln . By LC , let us denote the set of all crisp normal subgroups in C(X). Then LC is contained in Ln and so we state the following subdirect product theorem for the lattice of fuzzy normal subgroups. Theorem 9.6.1. Let (X,−1 , ∗) be a group. Then R|Ln : Ln → LJC is an isomorphism of the lattice Ln onto LI(X) that represents Ln as a subdirect product of the copies of LC in the lattice LJC .

256

9 Lattices of Fuzzy Subgroups

Proof. Since R is a bijection from FP(X) onto I(X), the function R|Ln is also a bijection from Ln onto LI(X). As in the subdirect theorem, the projection in each coordinate space LC is a surjection since for all r ∈ J and f ∈ LC , R(f )(r) = f. Also, Ln is closed under INF and SUP since for µ, ν ∈ Ln , µ∨ν = µ ∗ ν is a normal fuzzy subgroup. To show that R|Ln is an isomorphism, it remains to show that R|Ln commutes with INF and SUP. Since R commutes with inf by Proposition 9.1.4, R|Ln commutes with INF. That R|Ln commutes with SUP can be shown as follows: for µ, ν ∈ Ln we have µ ∨ ν = µ ∗ ν = µ ∨ ν . Thus for all r ∈ J, R(∨ν)(r) = R(µ ∨ ν )(r) = R(µ ∗ ν )(r) = R(µ )(r) ∗ R(ν )(r) = R(µ)(r) ∗ R(ν)(r) = R(µ)(r) ∨ R(ν)(r) = R(µ) ∨ R(ν)(r), where R(µ)(r) and R(ν)(r) are the tip extended pair of fuzzy subgroups associated with R(µ)(r) and R(ν)(r) respectively (for details see [15, 16]). Let C denote the set of all fuzzy congruences in the set of all fuzzy subsets FP(X ×X) and let C(X ×X) be the set of all crisp subsets of X ×X. Denote by CC the set of all crisp congruences in C(X × X). Then CC is a subset of C by [[8], Theorem 3.6]. Now consider the function R from the lattice FP(X × X) into the product lattice C(X × X)J . In view of the subdirect product theorem, the function R : FP(X × X) → C(X × X)J is a representation of the lattice FP(X × X) as a subdirect product of the copies of C(X × X) in the lattice C(X × X)J . In order to formulate the subdirect product theorem for the lattice C of fuzzy congruences in X, we denote by R|C the restriction of the function R : FP(X × X) → C(X × X)J to the lattice C and denote by CI(X × X) the image of R|C . Then we have the following subdirect product theorem for the lattice of fuzzy congruences of a group. J is an isoTheorem 9.6.2. Let (X,−1 , ∗) be a group. Then R|C : C → CC morphism of the lattice C onto CI(X × X) that represents C as a subdirect J . product of the copies of CC in the lattice CC

As a consequence of the above subdirect product theorems for the lattices of fuzzy normal subgroups and fuzzy congruences of a group, we next prove the isomorphism theorem. Theorem 9.6.3. Let G be a group. Then the lattice Ln of normal fuzzy subgroups of G is isomorphic with the lattice C of its fuzzy congruences.

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257

Proof. In view of subdirect product theorems for the lattices of fuzzy normal subgroups and fuzzy congruences, it suﬃces to show that the lattices LI(G) and CI(G × G) are isomorphic. Since the lattice LC of crisp normal subgroups is isomorphic to the lattice of crisp congruences CC, the product lattices LJC J J and CC are also isomorphic. The isomorphism θ : CC → LJC is given by J θ(A)(r) = N (A(r)), ∀A ∈ GC , ∀r ∈ J, where N (A(r)) is the fuzzy normal subgroup as deﬁned previously in expression 9.5.8. Since A(r) ∈ CC , it follows that N (A(r)) ∈ LC , ∀r ∈ J. It can be shown that under the restriction of θ to CI(G × G), we obtain the image LI(G). That is, the restriction of θ to CI(G × G) is a bijection onto LI(G) and hence is an isomorphism.

9.7 Fuzzy Subgroups With The Sup Property In this section, we examine fuzzy subgroups with the sup property. The results are essentially from [4]. We characterize this type of fuzzy subgroup in terms of their level subsets. We show that the property of being a fuzzy subgroup with the sup property is invariant under a homomorphism and under the homomorphic preimage of a fuzzy subgroup with the sup property. We consider the class Lnt of fuzzy normal subgroups of a group G, each of which assumes the same value “t” at the identity of G. It is proved that the subclass Lnst of fuzzy subgroups with sup property of Lnt constitutes a sublattice of Lnt . The modularity of Lnst follows as a consequence of the modularity of Lnt . It has been shown in [6] that the class of all fuzzy subgroups Lt , each of which assumes the same value t at the identity element of the given group, forms a complete sublattice of the lattice L of all fuzzy subgroups of that group. This class of fuzzy subgroups is known as the class of fuzzy subgroups with tip “t” and is shown to be of value in the study of fuzzy subgroups. For example, in [2], a fuzzy version of the correspondence theorem is obtained for this class of fuzzy subgroups. Moreover, in [1] and [6], it is established that in this class of fuzzy subgroups, the notion of set product due to Liu [20] can be used in the study of fuzzy subgroups as successfully as the notion of product of complexes in classical group theory. In [1], it was shown that in the class of fuzzy subgroups with tip “t” of a given group, if the set product of two fuzzy subgroups is a fuzzy subgroup, then it is the fuzzy subgroup generated by their union. This result leads us to the formation of lattices of fuzzy normal subgroups. In fact, it is proved in [6] that the class Lf nt of all fuzzy normal subgroups of a group, each of which has ﬁnite range and tip “t” is a modular sublattice of the lattice L of all fuzzy subgroups. On the other hand, it is shown in [8] that the condition of ﬁniteness of the range set of the fuzzy subgroups can be dropped. A diﬀerent technique is used here to establish this result. We prove that the class of all

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fuzzy normal subgroups with sup property and tip “t” of a group constitute a sublattice of L, which is modular. We recall some deﬁnitions and results from previous chapters. Let µ and ν be fuzzy subsets of a group G. Then the product µ ◦ ν is the fuzzy subset of G,deﬁned by for all z in G, µ ◦ ν(z) = ∨{µ(x) ∧ ν(x) | z = xy, x, y ∈ G}. A fuzzy subset µ in a set S is said to have sup property if for each nonempty subset A ⊆ S, there exists x0 ∈ A such that ∨{µ(x) | x ∈ A} = µ(x0 ). Let f be a homomorphism from a group G onto a group G , and let µ and ν be fuzzy subgroups of G and G , respectively. Then f (µ) and f −1 (ν) are fuzzy subgroups of G and G, respectively. The intersection of an arbitrary collection of fuzzy subgroups (normal fuzzy subgroups) of a group G is a fuzzy subgroup (normal fuzzy subgroup) of G. It is clear from the deﬁnition that any fuzzy subgroup of a ﬁnite group has the sup property. Moreover, a fuzzy subgroup with a ﬁnite image also possesses the sup property. We now provide an example of a fuzzy subgroup with the sup property whose image is countably inﬁnite. Example 9.7.1. Let G be the group of nonnegative real numbers less than 1, under the operation of addition modulo 1. Deﬁne the fuzzy subset µ of G as follows:  if x ∈ 1/2,   11 1 (1 + ) if x ∈ 1/2n+1 \1/2n ; n = 1, 2, ..., 2n µ(x) = 2 ∞  0 if x ∈ G\ ∪ 1/2n . n=1

Then µ is a fuzzy subgroup of G and its chain of level subgroups is given by: 1/2 ⊆ 1/22 ⊆ 1/23 ⊆ · · · ⊆ G.

∞

Let A be a nonempty subset of G. Then either A ⊆ G\ ∪ 1/2n or n=1

A ∩ 1/2n = ∅ for some positive integer n. Thus ∨{µ(x) | x ∈ A} = 0 = ∞

µ(x0 )∀x0 ∈ A if A ⊆ G\ ∪ 1/2n and otherwise ∨{µ(x) | x ∈ A} = 12 (1+ 21n ) n=1

= µ(x0 ) for the smallest such postive integer n such that A ∩ 1/2n = ∅ and where x0 ∈ A ∩ 1/2n . Hence µ has the sup property. We say that a fuzzy subgroup of µ of G attains its supremum everywhere on G if for each nonempty subset A ⊆ G, there exists x0 ∈ G such that ∨x∈A µ(x) = µ(x0 ). Example 9.7.2. Let Z be the group of integers under the binary operation of usual addition.  Deﬁne the fuzzy subset µ of Z as follows: if x ∈ Z\2, 0 if x ∈ (2n )\(2n+1 ), n = 1, 2, 3..., µ(x) = 12 (1 − 21n ) 1 if x = 0. 2 Then µ is a fuzzy subgroup of Z. Moreover, µ attains its supremum everywhere on Z. However, µ does not possess the sup property since µ fails to attain its supremum 1/2 on the subset Z\0.

9.7 Fuzzy Subgroups With The Sup Property

259

Example 9.7.3. In Example 9.7.2, deﬁne the fuzzy subset µ as given by  if x ∈ Z\2, 0 if x ∈ 2n \2n+1 , n = 1, 2, 3..., µ(x) = 12 (1 − 21n )  1 if x = 0. Then µ is a fuzzy subgroup of Z which neither attains its supremum everywhere nor has the sup property. The following theorem characterizes the notion of the sup property in terms of level subsets. Theorem 9.7.4. Let µ be a fuzzy subgroup of G. Then µ has the sup property if and only if µt ⊂ ∩ti
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Proof. By Theorem 1.2.10, f (µ) is a fuzzy subgroup of G . Let A be a subset of G . If f −1 (A) = φ, then the result is obvious. Suppse f −1 (A) = φ. Then ∨{f (µ)(y) | y ∈ A} = ∨{∨{µ(x) | x ∈ f −1 (y) | y ∈ A}} = ∨{µ(x) | −1 x ∈ f (y)}. Since µ has sup property, there exists x0 ∈ f −1 (A) such that ∨{µ(x)|x ∈ −1 f (A) = µ(x0 ). Hence y0 = f (x0 ) ∈ A and f (µ)(y0 ) = ∨{µ(x) | x ∈ f −1 (y0 )}. However, x0 ∈ f −1 (y0 ) ⊆ f −1 (A). Therefore, ∨{µ(x) | x ∈ f −1 (A)} ∨{µ(x) | x ∈ f −1 (y0 )}. Thus µ(x0 ) ∨{µ(x) | x ∈ f −1 (y0 )} µ(x0 ) since x ∈ f −1 (y0 ). Hence ∨µ(x) | x ∈ f −1 (y0 )} = µ(x0 ). Thus ∨{f (µ)(y) | y ∈ A} = µ(x0 ) = ∨{µ(x) | x ∈ f −1 (y0 )} = f (µ)(y0 ). Therefore, f (µ) has sup property. Theorem 9.7.6. Let f be a homomorphism from G into a group G and let ν be a fuzzy subgroup with sup property of G . Then the preimage f −1 (ν) is a fuzzy subgroup of G with the sup property. Proof. That f −1 (ν) is a fuzzy subgroup of G follows from Theorem 1.2.11. Let A be a subset of G. Then ∨{f −1 (ν)(x) | x ∈ A} = ∨{ν(f (x)) | x ∈ A} = ∨{ν(y) | y ∈ f (A)}. Since ν has sup property, there exists y0 ∈ f (A) such that ∨{ν(y) | y ∈ f (A)} = ν(y0 ). Hence ∨{f −1 (ν)(x) | x ∈ A} = ν(y0 ). Since y0 ∈ f (A), there exists x0 ∈ A such that y0 ∈ f (x0 ). Thus f −1 (ν)(x0 ) = ν(f (x0 )) = ν(y0 ). Therefore, f −1 (ν) has the sup property. The notion of the set product of fuzzy subsets was introduced in [20] as an extension of the notion of the product of complexes of classical group theory, and established some basic results to justify its appropriateness. The product of complexes plays an important role in the development of various aspects of group theory. The key to the use of this notion is that the product of two subgroups of a group is a subgroup if and only if the supbgroups commute. This fact leads to the important result that if the product of subgroups is a subgroup, then it is the least subgroup containing their union. In view of their importance in fuzzy group theory, we recall here some known results. Mashinchi and Zahedi [23, 24] have already obtained some of these results in more generality. Lemma 9.7.7. Let µ and ν be fuzzy subgroups of G. Then, µ ⊆ µ ◦ ν if and only if µ(e) ν(e).

9.7 Fuzzy Subgroups With The Sup Property

261

Proof. Let z ∈ G. Then µ ◦ ν(z) = ∨{µ(x) ∧ ν(y) | z = xy, x, y ∈ G} µ(z) ∧ ν(e) µ(z) ∧ µ(e) if µ(e) ν(e) = µ(z). Conversely, µ ◦ ν(e) = ∨{µ(x) ∧ ν(x−1 ) | x ∈ G} µ(e) ∧ ν(e). Thus if µ(e) > ν(e), then µ ◦ ν(e) = ν(e) < µ(e) and so µ µ ◦ ν.

Lemma 9.7.8. Let µ and ν be fuzzy subgroups of G. Then µ ⊆ µ ◦ ν and ν ⊆ µ ◦ ν if and only if µ(e) = ν(e). Proof. The proof follows by Lemma 9.7.7.

The next three results are restatements of Theorems 1.2.8, 1.2.9 and 1.3.1 respectively. Theorem 9.7.9. Let µ be a fuzzy subset of G. Then µ is a fuzzy subgroup of G if and only if (1) µ ◦ µ = µ, (2) µ(x) = µ(x−1 ) for x ∈ G. Theorem 9.7.10. Let µ and ν be fuzzy subgroups of G. Then µ ◦ ν = ν ◦ µ if either µ or ν is normal in G. Theorem 9.7.11. Let µ and ν be fuzzy subgroups of G. Then µ ◦ ν is a fuzzy subgroup of G if and only if µ ◦ ν = ν ◦ µ. The above theorem is established in a more general setting by Mashinchi and Zahedi in [24]. A similar generalization can be found in [[33], Theorem 5.1.10, p. 114]. Theorem 9.7.12. Let µ and ν be fuzzy subgroups of G such that µ(e) = ν(e). If µ ◦ ν is a fuzzy subgroup of G, then µ ◦ ν is generated by µ and ν. Proof. Suppose that µ ◦ ν is a fuzzy subgroup of G. Then we show that µ ◦ ν is the smallest fuzzy subgroup of G containing µ and ν. By Lemma 9.7.8, we see that µ ⊆ µ ◦ ν and ν ⊆ µ ◦ ν. Let θ be any fuzzy subgroup of G containing both µ and ν. Let z ∈ G. Then µ◦ν(z) = ∨{(µ(x)∧ ν(y)|z = xy, x, y ∈ G} ∨{θ(x)∧θ(y) | z = xy, x, y ∈ G} = θ ◦ θ(z) since µ(x) θ(x) and ν(y) θ(y). Since θ is a fuzzy subgroup of G, θ ◦ θ = θ by Theorem 9.7.9. Hence µ ◦ ν ⊆ θ. Consequently, µ ◦ ν is a fuzzy subgroup generated by µ and ν. > > Lemma 9.7.13. Let µ and η be fuzzy subgroups of G. Then (µ ◦ η)> t = µt µt for all t ∈ [0, 1].

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Proof. Let z ∈ G. Then z ∈ (µ ◦ ν)> t ⇔ µ ◦ ν(z) > t ⇔ ∨{µ(x) ∧ η(y) | z = xy, xy, ∈ G} > t ⇔ µ(x0 ) ∧ η(y0 ) > t for some x0, y0 ∈ G, z = x0 y0 > ⇔ x0 ∈ µ> t , y0 ∈ µt for some x0, y0 ∈ G, z = x0 y0 > > ⇔ z ∈ µt µt .

9.8 Lattices of Fuzzy Subgroups We now consider lattices of fuzzy subgroups. In this section, we continue to examine special types of lattices of fuzzy subgroups of a given group. It is shown in [6] that the class Lt of all fuzzy subgroups with tip “t” of a group G is a complete sublattice of the lattice L = L(G) of all its fuzzy subgroups. On the other hand, it is well known in classical group theory that the set of all normal subgroups of a group forms a modular sublattice of the lattice of its subgroups. It is this result which motivates the following discussion. Let t ∈ [0, 1]. We denote by Lnt , the class of all normal fuzzy subgroups of a group G with tip “t.” As usual, we denote by L the lattice of all fuzzy subgroups of G. The proof of the following result is similar to that of previous results such as Theorem 1.3.3. Theorem 9.8.1. Let µ be a fuzzy subgroup of G. Then the following statements are equivalent: (1) µ is normal. (2) µ> r is a normal subgroup of G for all r ∈ [0, t), t = µ(e). (3) µ> r is a normal subgroup of G for all r ∈ Imµ\{t}. (4) Every nonempty strong level subset µ> t is a normal subgroup of G. Theorem 9.8.2. Lnt is a sublattice of Lt and therefore of L. Proof. Let µ, ν ∈ Lnt . Then by Theorems 9.7.11 and 9.7.12, µ ◦ ν is the least fuzzy subgroup containing both µ and ν. Thus µ ∨ ν = µ ◦ ν. Clearly, µ ◦ ν(e) = t. In order to show that µ ∨ ν ∈ Lnt , it is suﬃcient to show that µ ◦ ν is normal. By Lemma 9.7.13, > > (µ ◦ ν)> r =(µ)r · (ν)r for all r ∈ [0, t). > By Theorem 9.8.1, (µ)> r and (ν)r are normal subgroups of G for each > r ∈ [0, t). Therefore, their product (µ)> r · (ν)r is a normal subgroup of G. > Hence (µ ◦ ν)r is a normal subgroup of G for all r ∈ [0, t). By Theorem 9.8.1, it follows that µ ◦ ν is a normal fuzzy subgroup of G. The intersection µ ∩ ν is a normal fuzzy subgroup of G by Theorem 1.4.6. In fact, µ ∩ ν is the largest fuzzy subgroup contained in µ and ν. Therefore, µ ∧ ν = µ ∩ ν ∈ Lnt . In order to construct the lattice of normal fuzzy subgroup with sup property, we begin with the following result.

9.8 Lattices of Fuzzy Subgroups

263

Theorem 9.8.3. Let µ and ν be fuzzy subgroups of G. If µ and ν have the sup property, then µ ∩ ν is a fuzzy subgroup of G with the sup property. Proof. It has been previously shown that µ ∩ ν is a fuzzy subgroup of G. Let A be a subset of G. Then ∨{µ ∩ ν(z) | z ∈ A} = ∨{µ(z) ∧ ν(z) | z ∈ A} = (∨{µ(z) | z ∈ XA }) ∨ (∨{ν(z) | z ∈ YA }), where XA = {z ∈ A | µ(z) ν(z)}, YA = {z ∈ A | ν(z) µ(z)}. Since µ and ν have the sup property, there exist x0 ∈ XA and y0 ∈ YA such that ∨{µ(z) | z ∈ XA } = µ(x0 ) and ∨{ν(z) | z ∈ YA } = ν(y0 ). Therefore, ∨{µ ∩ ν(z) | z ∈ A} = (µ(x0 ) ∨ ν(y0 ). Suppose that µ(x0 ) η(y0 ). Then ∨{µ ∩ ν(z) | z ∈ A} = µ(x0 ). Also, µ ∩ ν(x0 ) = µ(x0 ) ∧ η(x0 ) = µ(x0 ) since x0 ∈ XA . Hence µ ∩ ν attains its supremum at x0 and x0 ∈ XA ⊆ A. Suppose that µ(x0 ) ν(y0 ). Then, as in the previous case, it can be shown that ∨{µ ∩ ν(z) | z ∈ A} = ν(y0 ) = µ ∩ ν(y0 ) since y0 ∈ YA ⊆ A. Therefore, µ ∩ ν has the sup property. Theorem 9.8.4. Let µ and ν be fuzzy subgroups of G. If µ and ν have the sup property, then µ ◦ ν has the sup property. Proof. Let A ⊆ G. Then ∨{µ ◦ ν(z) | z ∈ A} = ∨{∨{µ(x) ∧ µ(y) | z = xy, x, y ∈ G | z ∈ A}}. Deﬁne the subsets X and Y of G × G as follows: X = {(x, y) ∈ G × G | µ(x) ν(y), z = xy and z ∈ A}, Y = {(x, y) ∈ G × G | ν(y) µ(x), z = xy and z ∈ A}. Then ∨{µ ◦ ν(z) | z ∈ A} = (∨{µ (x) ∧ ν (y) | (x, y) ∈ X}) ∨ (∨{µ(x) ∧ ν(y) | (x, y) ∈ Y }}) = (µ(x) | (x, y) ∈ X}) ∨ (∨{ν(y) | (x, y) ∈ Y }). Deﬁne the subsets Xl and Yr of G as follows: Xl = {x ∈ G | (x, y) ∈ X}, Yr = {y ∈ G | (x, y) ∈ Y }. Then ∨{µ ◦ ν(z) | z ∈ A} = (∨{µ(x) | x ∈ Xl }) ∨ (∨{ν(y) | y ∈ Yr }). Since µ and ν have the sup property, there exists x0 ∈ Xl and y0 ∈ Yr such that

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∨{µ(x) | x ∈ Xl } = µ (x0 ) and ∨{ν(y) | y ∈ Yr } = ν(y0 ). Therefore, ∨{µ ◦ ν(z) | z ∈ A} = µ(x0 ) ∨ ν(y0 ). Consider the cases µ(x0 ) ≥ ν(y0 ) and µ(x0 ) ν(y0 ). Suppose µ(x0 ) ν(y0 ). Then ∨{µ ◦ ν(z) | z ∈ A} = µ(x0 ). Since x0 ∈ Xl , there exists y0 ∈ G such that (x0, y0 ) ∈ X. Hence there exists z0 ∈ A such that z0 = x0 y0 and µ(x0 ) ν(y0 ). Choose such a y0 and consider z0 = x0 y0 ∈ A. We show that ∨{µ ◦ ν(z)|z ∈ A} = µ ◦ ν(z0 ). Deﬁne subsets X z0 = {(x, y) ∈ X | z0 = xy}, Y z0 = {(x, y) ∈ Y | z0 = xy}, and Xlz0 = {x ∈ G | (x, y) ∈ X z0 }, Yrz0 = {y ∈ G | (x, y) ∈ Y z0 }. Then µ ◦ ν(z0 ) = ∨{µ(x) ∧ ν(y) | z0 = xy, x, y ∈ G} = (∨{µ(x)∧ν(y) | (x, y) ∈ X z0 })∨(∨{µ(x)∧ν(y) | (x, y) ∈ Y z0 }) = (∨{µ(x) | x ∈ Xlz0 }) ∨ (∨{ν(y) | y ∈ Yrz0 }. z0 Since Xl ⊆ Xl and x0 ∈ Xlz0 , it follows that µ(x0 ) ∨{µ(x) | x ∈ Xlz0 } ∨{µ(x) | x ∈ Xl } = µ(x0 ). Hence ∨{µ(x) | x ∈ Xlz0 } = ∨{µ(x) | x ∈ Xl } = µ(x0 ). z0 Since Yr ⊆ Yr , we also have ∨{ν(y) | y ∈ Yrz0 } ∨{ν(y) | y ∈ Yr } = ν(y0 ). Thus ∨{ν(y) | y ∈ Yrz0 } ν(y0 ) µ(x0 ). Therefore, µ ◦ ν(z0 ) = µ(x0 )∨ (∨{ν(y) | y ∈ Yrz0 }) = µ(x0 ). Thus ∨{µ ◦ ν(z)|z ∈ A} = µ ◦ ν(z0 ). Hence µ ◦ ν attains its supremum at z0 ∈ A. Suppose µ(x0 ) ν(y0 ). Then ∨{µ ◦ ν(z) | z ∈ A} = ν(y0 ). Here it follows that ∨{µ ◦ ν(z) | z ∈ A} = µ ◦ ν(z0 ), where z0 ∈ A such that z0 = x0 y0 and ν(y0 ) µ(x0 ). Hence in both cases, µ ◦ ν attains its supremum at some element of A. Therefore, µ ◦ ν has the sup property. Let Lnst denote the set of all normal fuzzy subgroups of G with the sup property and with tip “t.” Theorem 9.8.5. Lnst is a sublattice of Lnt .

References

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Proof. Let µ, ν ∈ Lnst . Then µ, ν ∈ Lnt . Therefore, as in Theorem 9.8.2, µ ∨ ν = µ ◦ ν. Now by Theorem 9.8.4, µ ◦ ν has the sup property. Thus µ ∨ ν ∈ Lnst . Now µ ∧ ν = µ ∩ ν. Therefore, by Theorem 9.8.3, µ ∩ ν has the sup property. Hence, µ ∧ ν ∈ Lnst . Theorem 9.8.6. Lnst is a modular sublattice of Lt . Proof. The proof follows in a similar manner as that of Theorem 3.10 of [6]. Corollary 9.8.7. Lnst is a modular lattice. Proof. Lnst is modular since a sublattice of a modular lattice is modular.

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17. T. Head, Embedding lattices of fuzzy subgroups into lattices of crisp subgroups, Proceedings, Biennial Conference of the North American Fuzzy Information Proceedings Society NAFIPS’ 1996,184-186. 18. A. Jain, Tom Head’s join structure of fuzzy subgroups, Fuzzy Sets and Systems 125 (2002) 191-200. 19. N. Kuroki, Fuzzy Congruences and fuzzy normal subgroups, Inform. Sci. 60 (1992) 247-259. 253, 254 20. W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 21 (1982) 133-139. 257, 260 21. B. B. Makamba, Direct product and isomorphism of fuzzy subgroups, Inform. Sci. 65 (1992) 33-43. 22. B. B. Makamba and V. Murali, Normality and congruence in fuzzy subgroups, Inform. Sci. 59 (1992) 121-129. 253, 254 23. M. Mashinchi and M.M. Zahedi, Lattice structure of fuzzy subgroups, Bull. Iranian Math. Soc. 18 (2) (1992) 17-29. 247, 260 24. M. Mashinchi and M.M. Zahedi, On the product of T -fuzzy subgroups, Ann. Univ. Sci. Budapest. Sect. Comput., 12 (1991) 167-171. 247, 260, 261 25. N. P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225 - 239. 26. V. Murali, Fuzzy equivalence relations, Fuzzy Sets and Systems 30 (1989) 155163. 247 27. V. Murali, Fuzzy congruence relations, Fuzzy Sets and Systems 41 (1991) 359369. 247 28. V. Murali, Lattice of fuzzy subalgebras and closure systems in I, Fuzzy Sets and Systems 41 (1991) 101-111. 29. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. 30. N. Sultana and N. Ajmal, Generated fuzzy subgroup: a modiﬁcation, Fuzzy Sets and Systems 107 (1999) 241-243. 31. W. M. Wu, Normal fuzzy subgroups, Fuzzy Math., I(1) (1981) 21-30 (in Chinese). 247 32. W. M. Wu, Fuzzy congruences and normal fuzzy subgroups, Math. Appl., 1 (3) (1988) 9-20. 253, 254 33. Yu Yandong, J. N. Mordeson, S - C Cheng, Elements of L-algebra, Lecture Notes Vol. 1, Creighton University, Omaha, Nebraska, 1994. 261 34. L. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. 245

10 Membership Functions From Similarity Relations

Let G be a group and µ a fuzzy subgroup of G. Then µ can be thought to be the membership function of a fuzzy subgroup of G. In this chapter, we sometimes refer to µ in this way. We show that µ satisﬁes the equation µ(x) = σ(e, x), where σ is a similarity relation on G which is invariant under left-translation. We also show that under certain natural assumptions the elements x of G can be represented as permutations Px of a suitable universe Ω such that µ(x) equals the proportion of elements in Ω which are ﬁxed by Px . These results provide a deeper insight on the relationship of the group operation to the membership values µ(x). The membership values of a fuzzy subset of the universe X classify the elements of X into disjoint subsets Xt = {x ∈ X | µ(x) = t}. In many cases, the actual value taken by the membership function µ is not signiﬁcant in that one can replace the membership values t by f (t), where f is a strictly increasing function from [0, 1] onto [0, 1], and not change the basic properties of µ. This is still the case when there is a binary operation on X. Since any group can be viewed as a group of permutations of a suitable universe Ω (which can be taken to be G itself) with the identity element e representing the identity transformation on Ω, it is natural to deﬁne µ by comparing x with e in some way. We show that this is possible by considering similarity relations on Ω (or on G). The consideration of similarity relations is motivated by the fact that they satisfy a “min-transitivity” property that is similar to (1) of Deﬁnition 1.2.3. We then show that, under a natural assumption, the membership values µ(x) of a fuzzy subgroup can be viewed as the proportion of the elements of a suitable universe Ω which are the ﬁxed-points of Px , when every x ∈ G is represented as a permutation Px of Ω. For the sake of completeness and ease of reading, we occasionally restate and/or reprove results of previous chapters. When doing so, we try to give a diﬀerent approach and proof. The ﬁrst three sections are mainly from [7].

John Mordeson, Kiran R. Bhutani, Azriel Rosenfeld: Fuzzy Group Theory, StudFuzz 182, 267– 294 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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10.1 Similarity Relations and Membership Functions Let Ω be a set and σ be a fuzzy subset of Ω ×Ω. Then σ is called a similarity relation on Ω if the following properties properties hold: (S1) Reﬂexivity: σ(x, x) = 1 for all x ∈ Ω. (S2) Symmetry: σ(x, y) = σ(y, x) for all x, y ∈ Ω. (S3) Min-transitivity: σ(x, z) ≥ σ(x, y) ∧ σ(y, z) for all x, y, z ∈ Ω. Let σ be a similarity relation on Ω and G a group of permutations of Ω onto itself. Then it follows easily that σ deﬁned on G by Expression 10.1.1 below satisﬁes (S1)-(S3), and hence is a similarity relation on G. This construction is similar to that for deﬁning “distance” between two x and y of Ω into R : (x − y( = ∨{|x(w) − y(w)|; w ∈ Ω}. A general metric d(x(w), y(w)) on Ω can be used here rather than the absolute value |x(w) − y(w)|. Since the notion of “similarity” is opposite to that of “distance”, the inﬁmum is used in Expression 10.1.1 instead of the supremum. Let σ be the fuzzy subset of G × G deﬁned as follows: ∀x, y ∈ G, σ(x, y) = ∧{σ (x(w), y(w)) | w ∈ Ω}.

(10.1.1)

The min-transitivity property (S3) is veriﬁed as follows: ∀x, y, z ∈ G, σ(x, z) = ∧{σ (x(w), z(w)) | w ∈ Ω} ≥ ∧{σ (x(w), y(w)) ∧ σ (y(w), z(w)) | w ∈ Ω} ≥ (∧{σ (x(w), y(w)) | w ∈ Ω}) ∧ (∧{σ (y(w), z(w)) | w ∈ Ω}) = σ(x, y) ∧ σ(y, z). Let σ be a fuzzy subset of G. Then σ is said to be right-invariant if σ(x, y) = σ(xz, yz) for all x, y, z in G.

(10.1.2)

For all z ∈ G, {z(w) | w ∈ Ω} = Ω. Hence it follows that σ deﬁned by Expression (10.1.1) is right-invariant. The notion of left-invariance is deﬁned in a similar way. However, σ as deﬁned in Expression 10.1.1 may not be left-invariant. We say that σ is translation invariant if it is both left and right-invariant. Theorem 10.1.2 below shows that the converse of Expression 10.1.1 also holds. That is, each right-invariant similarity relation on a group is obtained in that way. First, we give an example to illustrate the construction in Expression 10.1.1. We make use of the following algorithm to determine the transitive closure of a fuzzy relation ρ on a set X. 1. ρ = ρ ∪ (ρ ◦ ρ). 2. If ρ = ρ, set ρ = ρ and go to step 1. 3. Stop ρ is the transitive max-min closure of ρ. Example 10.1.1. Let Ω = {1, 2, 3, 4}. Deﬁne the functions e, a, b, c of Ω onto itself as follows:

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269

e(1) = 1, e(2) = 2, e(3) = 3, e(4) = 4, a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 3, b(1) = 2, b(2) = 1, b(3) = 3, b(4) = 4, c(1) = 2, c(2) = 1, c(3) = 4, c(4) = 3. Then e, a, b, c are one-to-one functions of Ω onto itself. Under composition of functions, they form the Klein 4-group G = {e, a, b, c} with ab = ba = c and a2 = b2 = c2 = e. Deﬁne the fuzzy relation σ on Ω as follows: σ 1 2 3 4

1 1.0 0.2 0 0

2 3 0.2 0 1.0 0 0 1.0 0 0.3

4 0 0 0.3 1.0

Clearly, σ is reﬂexive and symmetric. A simple application of the algorithm preceding the example shows that σ is its transitive closure. Hence σ is a similarity relation on Ω. We now consider the corresponding similarity relation σ on G obtained by Expression 10.1.1. The values σ (i, j) and σ (j, i) for i ∈ {1, 2} and j ∈ {3, 4} do not have any inﬂuence here on the deﬁnition of σ. Now σ((x, y)) = ∧{σ (x(w), y(w)) | w ∈ Ω}. Thus σ((a, b)) = σ (a(1), b(1)) ∧ σ (a(2), b(2)) ∧ σ (a(3), b(3)) ∧ σ (a(4), b(4)) = σ ((1, 2)) ∧ σ ((2, 1)) ∧ σ ((4, 3)) ∧ σ ((3, 4)) = .2 ∧ .2 ∧ .3 ∧ .3 = .2 The remaining entries in the table given below are determined in a similar manner. σ e a b c e 1.0 0.3 0.2 0.2 a 0.3 1.0 0.2 0.2 b 0.2 0.2 1.0 0.3 c 0.2 0.2 0.3 1.0 Theorem 10.1.2. Let σ be a similarity relation on a group G. If σ is rightinvariant, then there is a universe Ω and a similarity relation σ on Ω such that G is isomorphic to a group of permutations of Ω onto itself and σ is obtained from σ as in Expression 10.1.1. Proof. Let Ω = G and let every x ∈ G correspond to the left translations Lx : G → G, where Lx (z) = xz ∀z ∈ G. The group product xy corresponds to the function composition Lx ◦ Ly (where Ly is applied ﬁrst) and the group inverse x−1 corresponds to the inverse transformation L−1 x where x, y ∈ G. Deﬁne the fuzzy subset σ of G × G by ∀(x, y) ∈ G × G, σ (x, y) = σ(x, y). Since σ (Lx (z), Ly (z)) = σ (xz, yz) = σ(xz, yz) = σ(x, y), the right-hand side of Expression 10.1.1 equals σ(x, y). Hence we regain σ from σ as desired.

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We note that if G is a group permutations on a given set Ω, then it may not be easy or even always possible to obtain a suitable σ on Ω which will give rise to the given σ via Expression 10.1.1. The next theorem shows the relationship between a right-invariant similarity relation on a group G and a fuzzy subgroup µ of G. Theorem 10.1.3. Let µ be a fuzzy subgroup of G such that µ(e) = 1. Then there exists a right-invariant similarity relation σ on G such that µ(x) = σ(e, x) ∀x ∈ G. Conversely, let σ be a right-invariant similarity relation on G. Then there exists a fuzzy group µ of G such that µ(x) = σ(e, x)∀x ∈ G. Proof. Suppose that µ is a fuzzy subgroup of G such that µ(e) = 1. Deﬁne the fuzzy subset σ of G × G by σ(x, y) = µ(xy −1 ) for all x, y ∈ G. Then σ(x, x) = µ(xx−1 ) = µ(e) = 1. Hence (S1) holds. Since µ(z) = µ(z −1 ) ∀z ∈ G, (S2) holds. For all x, y, z ∈ G, σ(x, z) = µ(xz −1 ) = µ(xy −1 yz −1 ) ≥ µ(xy −1 ) ∧ µ(yz −1 ) = σ(x, y) ∧ σ(y, z). Thus (S3) holds. Conversely, suppose σ is a right-invariant similarity relation on G. Since e has the highest membership value, it is natural to have a higher value of µ(x) if x is more similar to e according to σ. Deﬁne the fuzzy subset µ of G by µ(x) = σ(e, x)∀x ∈ G. Since σ is right-invariant, we have µ(x) = σ(e, x) = σ(ex−1 , xx−1 ) = σ(x−1 , e) = σ(e, x−1 ) by (S2) = µ(x−1 ). Now µ(xy) = σ(e, xy) ≥ σ(e, y) ∧ σ(y, xy) = σ(e, y) ∧ σ(e, x) = µ(y) ∧ µ(x). Thus µ is a fuzzy subgroup of G such that µ(e) = σ(e, e) = 1. Clearly, Theorem 10.1.3 holds if we replace the right-invariance of σ by left-invariance. In that case, given the fuzzy subgroup µ of G, we deﬁne the fuzzy subset σ of G × G by σ(x, y) = µ(x−1 y) for all x, y ∈ G. Hence σ(zx, zy) = µ(x−1 z −1 zy) = µ(x−1 y) = σ(x, y) for all z ∈ G. Suppose that σ is left-invariant. Deﬁne µ as before. Then ∀x, y ∈ G, µ(xy) = σ(e, xy) ≥ σ(e, x) ∧ σ(x, xy) = σ(e, x) ∧ σ(e, y) by left-invariance = µ(x) ∧ µ(y). Corollary 10.1.4. Let µ be a fuzzy subgroup of G and let σ be the similarity relation on G deﬁned by σ(x, y) = µ(xy −1 ) for all x, y ∈ G. Then σ is both left and right-invariant if and only if µ(xy) = µ(yx) for all x, y ∈ G.

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271

Proof. Suppose µ(xy) = µ(yx) for all x, y ∈ G. Then µ(x−1 y) = µ((x−1 y)−1 ) = µ(y −1 x) = µ(xy −1 ) for all x, y ∈ G. Thus it follows from the proof of Theorem 10.1.2 and the remarks following it that σ is both left and right-invariant. Conversely, suppose σ is left and right-invariant. Then it follows that for all x, y ∈ G, µ(xy) = µ((xy)−1 ) = σ(e, (xy)−1 ) = σ(e, y −1 x−1 ) = σ(y, x−1 ) by left-invariance = σ(yx, e) = σ(e, yx)

by right-invariance

= µ(yx). We recall that a fuzzy subgroup µ of a group G is called commutative if µ(xy) = µ(yx) for all x, y ∈ G. The equality µ(xy) = µ(yx) holds for certain pairs of elements x and y of G, namely those x, y ∈ G such that µ(x) = µ(y). This can be seen from the following result. Lemma 10.1.5. Let µ be a fuzzy subgroup of G. Let x, y ∈ G. If µ(x) = µ(y), then µ(xy) = µ(yx) = µ(x) ∧ µ(y). Proof. Suppose µ(x) > µ(y). Then µ(y) = µ(x−1 xy) ≥ µ(x−1 ) ∧ µ(xy) ≥ µ(x) ∧ µ(x) ∧ µ(y) = µ(y). Thus µ(x) ∧ µ(xy) = µ(y). Since µ(x) > µ(y), µ(xy) = µ(y) = µ(x) ∧ µ(y). Similarly, µ(yx) = µ(x) ∧ µ(y). The following result follows from Theorem 1.3.3. Lemma 10.1.6. Let µ be a fuzzy subgroup of G. Then µ(xy) = µ(yx) for all x, y ∈ G if and only if µt is a normal subgroup of G ∀t ∈ [0, µ(0)]. Suppose that µ is commutative. Then the similarity relation σ deﬁned by σ(x, y) = µ(xy −1 ) ∀x, y ∈ G is such that σ is a fuzzy subgroup of the group G × G. To see this, note that σ(x, y) = µ(xy −1 ) = µ(y −1 x) = µ((y −1 x)−1 ) = µ(x−1 y) = σ(x−1 , y −1 ) and that (x−1 , y −1 ) is the inverse of (x, y) in G × G∀x, y ∈ G. Moreover, σ((x, y)(x , y )) = σ(xx , yy ) = µ(xx (yy )−1 ) = µ(xx (y )−1 y −1 ) = µ(y −1 xx (y )−1 ) ≥ µ(y −1 x) ∧ µ(x (y )−1 ) = µ(xy −1 ) ∧ µ(x (y )−1 ) = σ(x, y) ∧ σ(x , y ). The converse is true as can be seen by the following argument. If the condition σ(x, y) = µ(xy −1 )∀x, y ∈ G implies σ is a fuzzy subgroup of G × G, then µ(xy) = µ(yx) for all x, y ∈ G. This follows since µ(xy) = σ(x, y −1 ) = σ(x−1 , y)∀x, y ∈ G by (G1) applied to G × G since (x−1 , y) is the inverse of (x, y −1 ). Finally, σ(x−1 , y) = µ(x−1 y −1 ) = µ((yx)−1 ) = µ(yx)∀x, y ∈ G.

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Given a fuzzy subgroup µ of G, we can construct a new fuzzy subgroup µ ¯ of G that satisﬁes µ ¯(xy) = µ ¯(yx) for all x, y ∈ G by using the observations made above. We ﬁrst construct a similarity relation σ ¯ from σ which is both left and right invariant as follows: ∀x, y ∈ G, σ ¯ (x, y) = ∧{σ(zx, zy) | z ∈ G}.

(10.1.3)

It follows easily that σ ¯ satisﬁes (S1)-(S3). We then deﬁne the fuzzy subset µ ¯ of G by ∀x ∈ G, µ ¯(x) = σ ¯ (e, x) = ∧{σ(z, zx) | z ∈ G} = ∧{σ(e, zxz −1 ) | −1 z ∈ G} = ∧{µ(zxz ) | z ∈ G}. It follows that µ ¯ is the largest fuzzy subgroup of G such that µ ¯ ⊆ µ and µ ¯ is commutative. The following theorem follows from these observations. Theorem 10.1.7. Let µ be a fuzzy subgroup of G. Then µ ¯ is the largest fuzzy subgroup of G such that µ ¯ ⊆ µ and µ ¯ is commutative, where µ ¯ is deﬁned by µ ¯(x) = ∧{µ(zxz −1 ) | z ∈ G} ∀x ∈ G. The following theorem follows from the observations made above and is easily proved directly as well from the fact that σ((x, y)(x , y )) = σ(xx , yy ) ≥ σ(xx , yx ) ∧ σ(yx , yy ) = σ(x, y) ∧ σ(x , y ) ∀x, y, x , y ∈ G. Theorem 10.1.8. Suppose σ is a translation invariant similarity relation on G. Then σ is a fuzzy subgroup of G × G if and only if µ is a fuzzy subgroup of G such that µ(x) = σ(e, x) and µ is commutative. If the similarity relation σ deﬁned by σ(x, y) = µ(xy −1 ) ∀x, y ∈ G is translation invariant, then there are two ways of embedding G and µ into G × G and σ. The mapping x → (e, x) preserves both the group operation and the membership values since µ(x) = σ(e, x) ∀x ∈ G. The same is true for the mapping x → (x, e). On the other hand, the mapping x → (x, x) preserves only the group operation, but not the membership values.

10.2 Level Subgroups, Cosets, and Equivalence Classes Let σ be a fuzzy relation on a set Ω. It follows that σ is a similarity relation on Ω if and only if σt = {(x, y) | σ(x, y) ≥ t} is an equivalence relation on Ω ∀ t ∈ [0, 1]. Lemma 10.2.1. Let µ be a fuzzy subgroup of G and let σ be the similarity relation on G deﬁned by σ(x, y) = µ(xy −1 ) (or, σ(x, y) = µ(x−1 y)) for all x, y ∈ G. Let [y]t = {x | σ(y, x) ≥ t}∀y ∈ G and ∀t ∈ [0, 1]. Then µt = [e]t Moreover, [y]t equals the right-coset (resp., left-coset) of y with respect to the subgroup µt .

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273

Proof. We have [y]t = {x | σ(y, x) ≥ t} = {x | σ(x, y) ≥ t} = {x | σ(e, xy −1 ) ≥ t} (resp., = {x | σ(e, y −1 x) ≥ 1}) = {x | xy −1 ∈ µt } (resp., = {x | y −1 x ∈ µt }) = µt y (resp., = yµt ).

In particular, [e]t = µt .

Example 10.2.2. Consider the similarity relation σ in Example 10.1.1 on G = {e, a, b, c} is µ(e) = 1, µ(a) = .3, µ(b) = µ(c) = .2, where µ(x) = σ(e, x)∀x ∈ G. There are three distinct level subgroups, namely, µ1 = {e} = µt for .3 < t ≤ 1, µ.3 = {e, a} = µt for .2 < t ≤ .3 for 0 ≤ t ≤ .2. µ.2 = G = µt For t = 1, there are 4 right-cosets {e}, {a}, {b}, and {c} of the levelsubgroup µ1 , which correspond to the equivalence-classes of σ1 = {(e, e), (a, a), (b, b), (c, c)}. For t = .3, there are 2 right-cosets {e, a} and {b, c} of µ.3 , which correspond to the equivalence classes of σ.3 = σ1 ∪ {(e, a), (a, e), (b, c), (c, b)}. Finally, for t = .2, there is only 1 right-coset {e, a, b, c} = G of µ.2 , which corresponds to the equivalence class of σ.2 = G × G. An important consequence of Lemma 10.2.1 is that each equivalence class determined by σt has the same size. As shown in the next theorem, this is all that can be said about σ as a fuzzy similarity relation on a ﬁnite group. In particular, the values taken by σ do not have any theoretical signiﬁcance (cf., Theorem 10.3.1). Theorem 10.2.3. Suppose σ is a fuzzy similarity relation on a ﬁnite set G. Then there is a binary operation on G such that G is a group and a fuzzy subgroup µ of G such that σ(x, y) = µ(xy −1 ) (or, = µ(x−1 y)) for all x, y ∈ G if and only if the equivalence classes determined by the crisp equivalence relation σt = {(x, y) | σ(x, y) ≥ t} have the same size for all t ∈ [0, 1]. Proof. Suppose the equivalence classes determined by σt are of the same size for all t ∈ [0, 1]. If σ(x, y) = 1 for all x, y ∈ G, then let µ(x) = 1 for all x ∈ G and any group operation of G, e.g., we can let G be a cyclic group of order |G|. Now assume that |G| = N and 0 ≤ t1 < t2 < · · · < tn = 1 are the distinct values of σ, where n ≥ 2. Let sj be the size of an equivalence class for σtj , j = 1, ..., n. Since σtj−1 ⊃ σtj as crisp equivalence relations, each equivalence class in σtj−1 is a disjoint union of equivalence classes of

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σtj , j = 2, ..., n. It follows that sn |sn−1 |sn−2 |...|s2 |s1 (= N ). Let Nj = N/sj , the number of equivalence classes in σtj . Clearly, (1 =) N1 |N2 |...|Nn−1 |Nn (≤ N ). We ﬁrst consider the cyclic group H generated by an element b of order N with the membership function (fuzzy subgroup) ν deﬁned on H as follows: ∀h ∈ H, for h = bkNn , k = 1, 2, ..., sn ν(h) = tn ν(h) = tn−1 for h = bkNn−1 , k = 1, 2, ..., sn−1 except for k being a multiple of Nn /Nn−1 ν(h) = tn−2 for h = bkNn−2 , k = 1, 2, ..., sn−2 except for k being a multiple of Nn−1 /Nn−2 ··· ··· for h = bkN1 , k = 1, 2, ..., s1 except for k being a multiple ν(h) = t1 of N2 /N1 . Clearly, the level subgroup νtj has the size sj for every j. By Lemma 10.2.1, it suﬃces to identify the elements of G with those H in a one-to-one and onto fashion so that for each t = tj , the equivalence class structures of σt coincide with that of the similarity relation obtained from ν(h) for h ∈ H. The following example shows that there may be more than one way to make the identiﬁcation alluded to in the proof of the previous theorem. Example 10.2.4. Let n = 3, N = 18 = s1 . Let s2 = 9 and s3 = 3. Then N1 = 1, N2 = 2, N3 = 6, and the following membership values on the cyclic group H of order 18 are as follows: ν(e) = ν(b6 ) = ν(b12 ) = t3 , ν(b2 ) = ν(b8 ) = ν(b10 ) = ν(b14 ) = ν(b16 ) = t2 , and the remaining ν(bj )’s equal t1 . The illustration below shows the equivalence classes of the similarity relation σ such that σ(x, y) = ν(xy −1 )∀x, y ∈ H. For an arbitrary set G of cardinality 18 and a similarity relation on G which takes 3 distinct values t1 < t2 < t3 = 1 such that the equivalence classes for the corresponding σtj have the equal size sj for each j = 1, 2, 3, there is a one-to-one function from G onto H to match the equivalence class structures in many ways. Any such function f determines a fuzzy subgroup µ of G such that µ(z) = ν(f (z))∀z ∈ G and moreover σ(z, z ) = µ(z(z )−1 )∀z, z ∈ G. We have the following equivalence classes. t3 -equivalence classes: {b, b7 , b13 }, {b3 , b9 , b15 }, {b5 , b11 , b17 }, {e, b6 , b12 }, {b2 , b8 , b14 }, {b4 , b10 , b16 } t2 -equivalence classes: {b, b7 , b13 }∪{b3 , b9 , b15 }∪{b5 , b11 , b17 }, {e, b6 , b12 }∪{b2 , b8 , b14 }∪{b4 , b10 , b16 } t1 -equivalence classes: {b, b7 , b13 }∪{b3 , b9 , b15 }∪{b5 , b11 , b17 }∪{e, b6 , b12 }∪{b2 , b8 , b14 }∪{b4 , b10 , b16 }.

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Corollary 10.2.5. Let µ be a fuzzy subset of a ﬁnite set G with the membership values 0 ≤ t1 < t2 < ... < tn = 1. Then there exists a group operation on G such that µ is a fuzzy group of G if and only if |µtj | divides |µtj−1 | for j = 2, ..., n. Proof. If G is a group, then |µtj | divides |µtj−1 | since µtj is a subgroup of µtj−1 , j = 2, ..., n. Suppose |µtj | divides |µtj−1 | for j = 2, ..., n. We decompose G into a series of disjoint subsets of equal size as shown in the above illustration. If t1 > 0, then we decompose the set G\µt1 arbitrarily into disjoint subsets each of size |µt1 |. If t1 = 0, then G = µt1 . We regard this decomposition as the equivalence class decomposition for a crisp equivalence relation σt1 . Next we decompose µt1 \µt2 into disjoint subsets of equal size |µt2 |, and also each of the other subsets of size |µt1 | obtained in the previous step are decomposed arbitrarily into disjoint subsets of size |µt2 |. This decomposition is taken to be the equivalence class decomposition for a crisp equivalence relation Rt2 ⊂ Rt1 . The process is continued until tn , deﬁning the equivalence class decomposition for the relation Rtn . We now deﬁne the similarity relation σ on G by σ(x, y) = ∨{tj | (x, y) ∈ Rtj }∀x, y ∈ G. An application of Theorem 10.2.3 yields the desired result. We now consider how the behavior of a fuzzy subset ν of a universe X compares with the result of Theorem 10.2.3. We must ﬁrst associate a suitable similarity relation ρ on X with ν. Deﬁne the fuzzy subset ρ of X ×X as follows: ∀x, y ∈ X, 1 if x = y ρ(x, y) = (10.2.1) ν(x) ∧ ν(y) otherwise. It follows easily that conditions (S1) − (S3) hold for ρ. If we assume that ν is normalized, that is, ν(z) = 1 for some z ∈ X, then we can recover ν from ρ simply by ν(x) = ρ(z, x). In general, we cannot recover ν from ρ as deﬁned by the above Expression 10.2.1. In particular, ∨{ρ(z, x) | z ∈ X} may not equal ν(x). Let Et = {(x, y) | ρ(x, y) ≥ t} denote the associated crisp equivalence relation. The equivalence classes [[y]]t with respect to Et are singleton sets if ν(y) < t, except possibly for one equivalence class which equals the level set νt if ν(y) ≥ t. This is illustrated in the Example 10.2.6. Example 10.2.6. Let X = {a, b, c, d} and let ν be the fuzzy subset of X deﬁned as follows: ν(a) = 1, ν(b) = 0.7, ν(c) = 0.3, and ν(d) = 0.2. . The similarity relation ρ, the level sets νt , and the equivalence classes [[y]]t are given as follows: ρ a a 1.0 b 0.7 c 0.3 d 0.2

b 0.7 1.0 0.3 0.2

c 0.3 0.3 1.0 0.2

d 0.2 0.2 0.2 1.0

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ν1 = {a} = νt , for 1.0 > t > 0.7 for 0.7 > t > 0.3 ν.7 = {a, b} = νt , ν.3 = {a, b, c} = νt , for 0.3 > t > 0.2 for 0.2 > t ≥ 0.0 ν.2 = X = νt , For For For For

0.7 < t ≤ 1.0 : {a} = νt , {b}, {c}, {d}; 4 classes 3 classes 0.3 < t ≤ 0.7 : {a, b} = νt , {c}, {d}; 2 classes 0.2 < t ≤ 0.3 : {a, b, c} = νt , {d}; 1 class 0.0 ≤ t ≤ 0.3 : {a, b, c, d} = νt .

All equivalence classes [[y]]t are singleton sets except possibly for one which equals the level set νt . Suppose G is an Abelian group. Then unlike σ, ρ may not always be a fuzzy subgroup of G × G. Clearly, ρ(x−1 , x−1 ) = 1 = ρ(x, x) and for x = y, ρ(x−1 , y −1 ) = µ(x−1 ) ∧ µ(y −1 ) = µ(x) ∧ µ(y) = ρ(x, y). However, consider the membership function µ(e) = 1.0, µ(a) = 0.3, µ(b) = 0.2 = µ(c) obtained from σ in Example 10.1.1, and apply the construction for ρ to µ. We have ρ(b, b) = 1 and ρ(e, a) = µ(e) ∧ µ(a) = 0.3, but ρ((e, a)(b, b)) = ρ(eb, c) = µ(b) ∧ µ(c) = 0.2 < ρ(e, a) ∧ ρ(b, b). It is not surprising that ρ fails to be a fuzzy subgroup of G × G since Expression 10.2.1 does not make use of the group property of G. Theorem 10.2.7. Let G be an Abelian group and let µ be a fuzzy subgroup of G. Then σ deﬁned by σ(x, y) = µ(xy −1 ) ∀x, y ∈ G is the smallest similarity relation containing ρ as deﬁned in the Expression 10.2.1 with ν replaced by µ and such that σ is a fuzzy subgroup of G × G. Proof. Suppose that σ is a similarity relation on G such that σ ⊇ ρ, where ρ is a similarity relation on G such that σ is a fuzzy subgroup of G × G. Then σ(x, y) = σ((xy −1 , e)(y, y)) ≥ σ(xy −1 , e) ∧ σ(y, y) ≥ ρ(xy −1 , e) ∧ ρ(y, y) = µ(xy −1 ) ∧ µ(e) and 1 = µ(xy −1 ). Since σ(x, y) = µ(xy −1 ) ∀x, y ∈ G, σ is a fuzzy subgroup of G × G.

10.3 Representation of Membership Functions In the previous sections, we considered similarity relations on a group G, their invariant properties, and their relationship to fuzzy subgroups of G. We now consider to what extent the membership function µ can represent realistic properties of the group elements. As seen from Theorem 10.2.3 and Corollary 10.2.5 for a ﬁnite group, the values tj taken by µ have no direct role except for separating the elements into disjoint equivalence classes for the associated similarity relation σ. If we regard each x ∈ G as a 1-1 onto function Px of a set Ω onto itself and we attempt to deﬁne µ in a direct way by looking at the properties of Px , then it is not always easy to satisfy the property that

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µ(xy) ≥ µ(x) ∧ µ(y). For example, suppose Ω is ﬁnite. Then deﬁne µ in terms of the set of ﬁxed points of Px as follows: ∀x ∈ Ω, µ(x) = |φ(x)|/|Ω|, where φ : Ω → P(Ω) maps φ(x) = {w ∈ Ω | Px (w) = w}. (10.3.1) In general, we can only say that µ(xy) ≥ 0 ∨ (µ(x) + µ(y) − 1), which is weaker than the property that µ(xy) ≥ µ(x)∧µ(y), even though the properties µ(e) = 1 and µ(x) = µ(x−1 ) hold, where x, y ∈ G. For the mappings b and c in Example 10.1.1, we have µ(c) = 0 and µ(a) = 2/4 = µ(b) and so µ satisﬁes the above Expression 10.3.1, but µ(ab) µ(a) ∧ µ(b). However, if we assume that for all x, y ∈ G either φ(x) contains φ(y) or φ(y) contains φ(x), then the property µ(xy) ≥ µ(x)∧µ(y) holds since φ(xy) ⊇ φ(x) ∩ φ(y). We show below that, under some natural assumptions, there is a representation of the elements of G as one-to-one functions of a suitable universe Ω onto itself such that the property φ(x) ⊇ φ(y) or φ(y) ⊇ φ(x) holds for all x, y ∈ G. Note that the similarity relation σ associated with µ given by Expression 10.3.1 is such that σ(x, y) equals |{w | Px (w) = Py (w)}|/|Ω|, which is the proportion of the elements in Ω, where the mappings associated with x and y agree. Suppose µ(x) = 1 for some x = e. Then Px = Pe , the identity mapping on Ω. Thus the association x → Px is not a group isomorphism (from G to the group of one-to-one functions of Ω onto itself). Henceforth, we assume that µ takes at least two distinct values. We say µ satisﬁes the identity property if µ(x) < 1 = µ(e) for all x = e. Theorem 10.3.1. Suppose µ is a fuzzy subgroup of a ﬁnite group G such that µ(e) = 1. Let 0 ≤ t1 < t2 < ... < tn = 1 be the membership values of µ with n ≥ 2. Then there exists a representation of G involving permutations Px on a suitable universe Ω such that Expression 10.3.1 holds if and only if each tj is a rational number and µ(xy) = µ(yx) for all x and y. The permutations Px are distinct if µ has the identity property. The only-if part in Theorem 10.3.1 follows easily since given the mappings Px , φ(yxy −1 ) = {Py (w) | w ∈ φ(x)} and so |φ(x)| = |φ(yxy −1 )| for all x, y ∈ G. Thus µ(x) = µ(yxy −1 ) and so µ(xy) = µ(yx) for all x, y ∈ G. By Expression 10.3.1, each tj or µ(x) is a rational number. Before we prove the if-part of the theorem, we need the following lemma. Lemma 10.3.2. Suppose µ is a fuzzy subgroup of G and µt is a normal subgroup of G for some t ∈ µ(G). Consider the quotient group G/µt . Then µ is a fuzzy subgroup of G/µt with the identity property, where µ is deﬁned on the cosets of G/µt as follows: ∀[x] ∈ G/µt , 1 if x ∈ µt , (10.3.2) µ ([x]) = µ(x) otherwise. Moreover, if G is commutative so is µ .

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Proof. We ﬁrst show that µ is well-deﬁned. Suppose [x] = [x ] (or equivalently, x = xz for some z ∈ µt ). Then µ ([x]) = µ ([x ]). If both x and x ∈ µt , then there is nothing to prove. If both x and x don’t belong to µt , then µ(x ) ≥ µ(x) ∧ µ(z) = µ(x). Similarly from x = x z −1 , where z −1 ∈ µt , we have µ(x) ≥ µ(x ) and hence µ(x) = µ(x ). That µ ([x]) = µ ([x]−1 )∀x ∈ G follows immediately. To prove that µ ([x][y]) ≥ µ ([x]) ∧ µ ([y]), we only need / µt and hence to consider the case µ ([x][y]) = µ ([xy]) < 1. In that case, xy ∈ / µt and µ ([xy]) = µ(xy). If both x and y do not belong to µt , yx = yxyy −1 ∈ then µ ([xy]) = µ(xy) ≥ µ(x) ∧ (y) = µ ([x]) ∧ µ ([y]) and we have the desired / µt and hence µ ([x]) ∧ µ ([y]) = 1 ∧ µ(y) = µ(y). result. If x ∈ µt , then y ∈ Also, since µt is a normal subgroup of G, [xy] = xyµt = µt xy = µt y = yµt = [y]. Thus µ(xy) = µ(y) and so µ ([x][y]) = µ ([x])∧µ ([y]). A similar argument applies if x ∈ / µt and y ∈ µt . That µ is commutative follows from the fact that xy ∈ µt if and only if yx ∈ µt . That µ satisﬁes the identity property is immediate. Proof of Theorem 10.3.1 We may assume without loss of generality that µ has the identity property. Otherwise, let G = G/µ1 and let [x] denote the coset of x with respect to µ1 . By Lemma 10.3.2, µ ([x]) = µ(x) and so µ is a fuzzy subgroup of G , where µ is commutative and has the identity property. Also, Im(µ ) = Im(µ). Assume that there is a suitable representation for the elements of G as in Theorem 10.3.1, where P[x] is the mapping associated with [x]. We immediately obtain a desired representation of G by lifting the representation of G via the mapping x → [x] from G to G , i.e., we associate P[x] with x. We prove the theorem by induction on the number n ≥ 2 of distinct values of µ. Suppose that n = 2. If t1 = 0, then we let Ω = G and let Px be the lefttranslation Px (z) = Lx (z) = xz from Ω onto Ω. This gives φ(x) = ∅ for x = e and φ(e) = Ω. Thus Expression 10.3.1 is satisﬁed and clearly Pxy = Px ◦ Py . The Px ’s are distinct since |G| ≥ 2. If t1 = p/q > 0, we take Ω to be q copies of the set G. We deﬁne Px as follows. For each of the ﬁrst q − p copies of G, we deﬁne Px to be the same as Lx and for each of the remaining p copies of G, we deﬁne Px to be the identity map. Once again the mappings Px satisfy expression (10.3.1), are distinct, and Pxy = Px ◦ Py . Now assume that the theorem is true for n = m (≥ 2). We prove that it is true for n = m + 1. Consider the following two fuzzy subsets of G deﬁned as follows: ∀x ∈ G  if x = e, 1 µ1 (x) = tn−2 if µ(x) = tn−1 , (10.3.3)  µ(x) otherwise;  if x = e, 1 if µ(x) = tn−1 , µ2 (x) = 1 (10.3.4)  µ(x) otherwise.

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Both µ1 and µ2 take only m distinct values {t1 , t2 , ..., tn−2 , 1}. Since (µi )t = {x | µi (x) ≥ t} is clearly a subgroup of G for all t ≥ 0 and i = 1, 2, both µ1 and µ2 are fuzzy subgroups of G. Also, both µ1 and µ2 are commutative, but only µ1 has the identity property. By the induction hypothesis, for each i = 1, 2, (i) we have a representation of µi as 1-1 functions Px of a ﬁnite domain Ωi onto itself such that µi (x) = |φi (x)|/|Ωi |, where φi (x) is the set of ﬁxed points of (i) (i) Px in Ωi , i = 1, 2. Moreover, the functions Px are distinct. Let N1 and N2 be integers such that N1 tn−2 |Ω1 | + N2 |Ω2 | = tn−1 (N1 |Ω1 | + N2 |Ω2 |). Since tn−2 < tn−1 < 1, this is always possible. Let Ω consist of N1 copies of Ω1 and (i) N2 copies of Ω2 . We copy the functions Px on each copy of Ωi . For all x ∈ G, we then have |φ(x)| = N1 |φ1 (x)| + N2 |φ2 (x)|. In particular, if µ(x) = tj , j ≤ n − 2, then |φ(x)| = N1 tj |Ω1 | + N2 tj |Ω2 | = tj |Ω|. Clearly, µ(e) = 1. Finally, if µ(x) = tn−1 , then |φ(x)| = N1 tn−2 |Ω1 | + N2 |Ω2 | = tn−1 |Ω|, where x ∈ G. The following example illustrates the construction in the previous theorem. Example 10.3.3. We illustrate the construction in the proof of Theorem 10.3.1 using the group and fuzzy subgroup considered in Example 10.2.2, namely, G = {e, a, b, c}, where c = ab = ba and a2 = b2 = e, and the membership function given by µ(e) = 1, µ(a) = 0.3, and µ(b) = 0.2 = µ(c). The left translations Lx (z) = xz on G are as follows: Le (e) = e, Le (a) = a, Le (b) = b, Le (c) = c, La (e) = a, La (a) = e, La (b) = c, La (c) = b, Lb (e) = b, Lb (a) = c, Lb (b) = e, Lb (c) = a, Lc (e) = c, Lc (a) = b, Lc (b) = a, Lc (c) = e. The representations of G based on the fuzzy subgroups µ1 and µ2 respectively, given by Expression 10.3.3 can be determined. The ﬁnal representation is obtained by choosing N1 = 7 copies of (i) below and N2 = 2 copies of (ii) below. This gives |Ω| = 7(4 × 4 + 4) + 2(2 × 4 + 2) = 160 and |φ(b)| = 2 × 10 + 7 × 4 = 48. The proportion of ﬁxed-points of Pa equals 48/160 = 0.3 = µ(a), as desired. Similarly, the proportion of ﬁxed-points of both Pb and Pc equal 0.2. We have 4 copies of the following: Pe (e) = e, Pe (a) = a, Pe (b) = b, Pe (c) = c, Pa (e) = a, Pa (a) = e, Pa (b) = c, Pa (c) = b, Pb (e) = b, Pb (a) = c, Pb (b) = e, Pb (c) = a, Pc (e) = c, Pc (a) = b, Pc (b) = a, Pc (c) = e. We have 1 copy of the following: Px (z) = z ∀x, z ∈ {e, a, b, c}. (i) We have 4 copies of the following: Pe ([e]) = [e], Pe ([b]) = [b], Pa ([e]) = [e], Pa ([b]) = [b], Pb ([e]) = [b], Pb ([b]) = [e], Pc ([e]) = [b], Pc ([b]) = [e], where [e] = [a], [b] = [c].

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We have 1 copy of the following: Pe ([e]) = [e], Pe ([b]) = [b], Pa ([e]) = [e], Pa ([b]) = [b], Pb ([e]) = [e], Pb ([b]) = [b], Pc ([e]) = [e], Pc ([b]) = [b], where [e] = [a], [b] = [c]. (ii) (i) gives the representation of the group G = {e, a, b, c} for the membership function µ1 (e) = 1 and µ1 (a) = µ1 (b) = µ1 (c) = 0.2. (ii) gives the representation of G, based on that of the quotient group G/(µ2 ).3 = {e, a, b, c}/{e, a}, where µ2 (e) = 1 = µ2 (a) and µ2 (b) = 0.2 = µ2 (c).

10.4 Fuzzy Subgroups Based on Group Properties In this section, an algebraic approach to the construction of fuzzy subgroups is discussed [16]. Let L = (L, ∨, ∧, 0, 1) be a complete lattice. An L-fuzzy subset of a set X is a function of X into L. The approach in this section is the construction of L-fuzzy subgroups µ of a group G so that for each x ∈ G, the lattice element µ(x) signiﬁes the membership grade of x and µ(x) is determined by the extent to which x satisﬁes some algebraic property. Deﬁnition 10.4.1. A complete Heyting algebra H is a complete lattice (H, ∨, ∧, 0, 1) such that ∀a ∈ H and ∀B ⊆ H, ∨{a ∧ b | b ∈ B} = a ∧ (∨B). Let I = [0, 1]. Then I = (I, ∨, ∧, 0, 1) and P(S) = (P(S), ∪, ∩, ∅, S) are complete Heyting algebras, where S is a nonempty set and P(S) is the power set of S. We adopt the usual convention that ∨∅ = 0 and ∧∅ = 1 for I and ∨∅ = ∅ and ∧∅ = S for P(S). Deﬁnition 10.4.2. Let H be a complete Heyting algebra and let G be a group. An H-fuzzy subset µ of G is called an H-fuzzy subgroup of G if the following conditions hold: (1) µ(e) = 1, where e is the identity of G, (2) µ(xy) ≥ µ(x) ∧ µ(y) for all x, y ∈ G, (3) µ(x−1 ) ≥ µ(x) for all x ∈ G. Example 10.4.3. Let G be an (additive) Abelian group and let nG = {nx | x ∈ G}, where n is a positive integer. Let m be a positive integer. Let µ be the I-fuzzy subset of G deﬁned as follows: ∀x ∈ G, µ(x) = ∨{1 − 2−k | k ∈ N and x ∈ mk G}. Then µ(0) = 1. Note that µ measures the membership grade of x in µ by the degree to which x is divisible by m. The higher the power of m dividing x, the greater the degree of membership of x.We now show that µ is an I-fuzzy subgroup of G. Let x, y ∈ G. Suppose x ∈ mp G and y ∈ mq G. then x + y ∈ mp∧q G. Thus µ(x + y) ≥ 1 − 2p∧q = (1 − 2−p ) ∧ (1 − 2−q ). Hence µ(x + y) ≥ ∨{(1 − 2−p ) ∧ (1 − 2−q ) | p, q ∈ N } = µ(x) ∧ µ(y). Since mk G is a group for all positive integers m and for all k ∈ N, it follows that x ∈ mk G if and only if x−1 ∈ mk G. Thus it follows that µ(x) = µ(x−1 ). Hence µ is an I-fuzzy subgroup of G.

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Example 10.4.4. Let G be an (additive) Abelian group and let m be an integer. Deﬁne the P(N)-fuzzy subset µ of G as follows: ∀x ∈ G, µ(x) = {k ∈ N | x ∈ mk G}. Then µ(0) = N. Let x, y ∈ G. Since x ∈ mk G if and only if −x ∈ mk G, µ(x) = µ(−x). Now x ∈ mk G and y ∈ mj G imply x+y ∈ mk∧j G. Thus µ(x + y) ⊇ µ(x) ∩ µ(y). Hence it follows that µ is a P(N)-fuzzy subgroup of G. Example 10.4.5. Let G be an (additive) Abelian group and let Z+ denote the positive integers. Deﬁne the P(Z+ )-fuzzy subset µ of G as follows: ∀x ∈ G, µ(x) = {n ∈ Z+ | x ∈ nG}. Then µ(0) = Z+ . Let x, y ∈ G. Then x ∈ nG if and only if −x ∈ nG. Thus µ(x) = µ(−x). Now x ∈ nG and y ∈ mG imply x + y ∈ (n ∧ m)G. Hence µ(x + y) ⊇ µ(x) ∩ µ(y). Thus it follows that µ is an P(Z+ )-fuzzy subgroup of G. It is clear that G is divisible if and only if µ is a constant on G. Example 10.4.6. Let P denote the set of all primes in Z. Deﬁne the P(P )fuzzy subset µ of Z as follows: ∀n ∈ Z, µ(n) = {p ∈ P | p divides n}. Then µ(0) = P . Let n, m ∈ Z. Then p divides n if and only if p divides −n. Thus µ(n) = µ(−n). Also, p divides n and p divides m imply p divides n+m. Hence µ(n + m) ⊇ µ(n) ∩ µ(m). Thus µ is a P(P )-fuzzy subgroup of Z. (In fact, µ is a fuzzy subring of Z.) We can think of µ as measuring each n = ±1 by the extent to which n is a composite. Example 10.4.7. Let K[x] denote the ring of polynomials over the ﬁeld K. Deﬁne the P(K)-fuzzy subset µ of K[x] as follows: ∀p(x) ∈ K[x], µ(p(x)) = {k ∈ K | p(k) = 0}. Then µ(0) = K. Let p(x), q(x) ∈ K[x]. Then p(k) = 0 if and only if −p(k) = 0. Thus µ(p(x)) = 0 = µ(−p(x)). Suppose p(k) = 0 and q(k) = 0. Then p(k)+q(k) = 0. Hence µ(p(x)+q(x)) ⊇ µ(p(x))∩µ(q(x)). Thus µ is a P(K)-fuzzy subgroup of K[x]. (In fact, µ is a P(K)-fuzzy subring and P(K)-fuzzy subspace of K[x].) We can think of µ as measuring the polynomial by the set of its roots in K. If p(x) is a constant polynomial, but not 0, then µ(p(x)) = ∅, while µ(p(x)) = K if p(x)) = 0. Example 10.4.8. Let K be a ﬁeld and Mat n [K] denote the set of all n × n matrices over K, where n ∈ N with n > 1. Let K n denote the set of all ordered n-tuples with entries from K. Deﬁne the P(K n )-fuzzy subset µ of Mat n [K] as follows: ∀A ∈ Mat n [K], µ(A) = {x ∈ K n | Ax = kx for some k ∈ K}. Then µ()) = K n , where ) denotes the zero matrix. Let A, B ∈ Mat n [K]. Then Ax = kx if and only if (−A)x = (−k)x. Thus µ(A) = µ(−A). Now Ax = k1 x and Bx = k2 x imply (A + B)x = (k1 + k2 )x. Hence µ(A + B) ⊇ µ(A) ∩ µ(B). Thus µ is a P(K n )-fuzzy subgroup of Mat n [K]. (In fact, µ is a P(K n )-fuzzy subring and a P(K n )-fuzzy subspace of Mat n [K].) We can think of µ as measuring A by its set of eigenvectors. Example 10.4.9. Let G and H be groups and let Hom(G, H) denote the set of all homomorphisms of G into H. Let S be a subgroup of H. Deﬁne the P(Hom(G, H))-fuzzy subset µ of G as follows: ∀x ∈ G, µ(x) = {f ∈

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Hom(G, H) | f (x) ∈ S}. Then µ(e) = Hom(G, H). Let x, y ∈ G. Then f (x) ∈ S if and only if f (−x) ∈ S. Thus µ(x) = µ(−x). Now f (x), f (y) ∈ S implies f (xy) = f (x)f (y) ∈ S. Hence µ(xy) ⊇ µ(x) ∩ µ(y). Then µ is a P(Hom(G, H))-fuzzy subgroup of G. Example 10.4.10. Let G be a group and H be (additive) Abelian group. Let Hom(G, H) denote the set of all homomorphisms of G into H. Let S be a subgroup of H. Now Hom(G, H) is a an Abelian group under the operation of pointwise addition of homomorphisms. Deﬁne the P(G)-fuzzy subset µ of Hom(G, H) as follows: ∀f ∈ Hom(G, H), µ(f ) = {x ∈ G | f (x) ∈ S}. Then µ()) = S, where )(x) = 0 ∀x ∈ G. Let f, g ∈ Hom(G, H). Then f (x) ∈ S if and only if (−f )(x) = −f (x) ∈ S. Thus µ(f ) = µ(−f ). Now f (x), g(x) ∈ S imply (f + g)(x) = f (x) + g(x) ∈ S. Hence µ(f + g) ⊇ µ(f ) ∩ µ(g). Thus µ is a P(G)-fuzzy subgrouip of Hom(G, H). If S consists of just the zero element, then µ can be thought of measuring f by the size of its kernel. Example 10.4.11. Let X be a set and SX be the symmetric group on X, i.e., the group of all permutations of X. Deﬁne the P(X)-fuzzy subset µ of SX as follows: ∀f ∈ SX , µ(f ) = {x ∈ X | f (x) = x}. Then µ(i) = X, where i is the identity function on X. Let f, g ∈ SX . Then f (x) = x if and only if f −1 (x) = x. Thus µ(f ) = µ(f −1 ). Now f (x) = x and g(x) = x imply (g ◦ f )(x) = g(f (x)) = g(x) = x. Hence µ(g ◦ f ) ⊇ µ(g) ∩ µ(f ). Thus µ is a P(X)-fuzzy subgroup of SX . We can think of µ as assigning higher degrees of membership to those permutations that ﬁx larger subsets of X. Example 10.4.12. Let X = {1, 2, ..., n} be the vertices of a regular n-gon. Let Dn denote the dihedral group of all rigid motions of the n-gon that leave the n-gon coincident with itself. Deﬁne the P(X)-fuzzy subset µ of Dn as follows: ∀f ∈ Dn , µ(f ) = {x ∈ X | f (x) = x}. Then µ(i) = X, where i denotes the identity of Dn . Let f, g ∈ Dn . Then f (x) = x if and only if f −1 (x) = x. Thus µ(f ) = µ(f −1 ). Now f (x) = x and g(x) = x imply (g ◦ f )(x) = x. Hence µ(g ◦ f ) ⊇ µ(g) ∩ µ(g). Thus µ is P (X)-fuzzy subgroup of Dn . It is natural to call µ the fuzzy dihedral subgroup Dn . Example 10.4.13. Let H be a group and let G be a group acting on H. That is, there is a homomorphism f : G →Aut(H), where Aut(H) denotes the group of all automorphisms of H. Denote f (x)(y) by y x , where x ∈ G and y ∈ H. Deﬁne the P (G)-fuzzy subset µ of H as follows: ∀y ∈ H, µ(y) = {x ∈ G | y x = y}. Then µ(e) = G. Let y ∈ H. Then f (x)(y) = y if and only if f (x)(y −1 ) = y −1 . Thus µ(y) = µ(y −1 ). Let y1 , y2 ∈ H. Then f (x)(y1 ) = y1 and f (x)(y2 ) = y2 imply f (x)(y1 y2 ) = f (x)(y1 )f (x)(y2 ) = y1 y2 . Hence µ(y1 y2 ) ⊇ µ(y1 ) ∩ µ(y2 ). Thus µ is a PG)-fuzzy subgroup of H. We can think of µ as measuring the degree of membership of an element h by the size of its isotropic subgroup. Special cases of this can also be considered. For example, the action of a group on itself by right translations or the action of a group on a normal subgroup by conjugation. It is natural to call this class of fuzzy subgroups as the class of isotropic fuzzy subgroups.

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Example 10.4.14. Let H be a group and let G be a group acting on H. Let S be a subgroup of G. Deﬁne the P(G)-fuzzy subset µ of H as follows: ∀y ∈ H, µ(y) = {x ∈ G | y x ∈ S}. Then µ(e) = G. Let y ∈ H. Then f (x)(y) ∈ S if and only if f (x)(y −1 ) ∈ S. Thus µ(y) = µ(y −1 ). Let y1 , y2 ∈ H. Then f (x)(y1 ) ∈ S and f (x)(y2 ) ∈ S imply f (x)(y1 y2 ) = f (x)(y1 )f (x)(y2 ) ∈ S. Hence µ(y1 y2 ) ⊇ µ(y1 ) ∩ µ(y2 ). Thus µ is a P(G)-fuzzy subgroup of H. Example 10.4.15. Let G be a group. For all x ∈ G, let C(x) denote the centralizer of x in G. Deﬁne the P(G)-fuzzy subset µ of G as follows: ∀x ∈ G, µ(x) = C(x)∩C(x−1 ). Then µ(e) = G. Let x, y ∈ G. Since C(x)∩C(x−1 ) = C(x−1 ) ∩ C(x), µ(x) = µ(x−1 ). Since ∀z ∈ G, zx = xz, zx−1 = x−1 z, zy = yz, zy −1 = y −1 z imply (xy)z = z(xy) and (xy)−1 z = z(xy)−1 , we have that C(xy) ∩ C((xy)−1 ) ⊇ C(x) ∩ C(x−1 ) ∩ C(y) ∩ C(y −1 ). Hence µ(xy) ⊇ µ(x) ∩ µ(y). Thus µ is a P(G)-fuzzy subgroup of G. We can think of µ as measuring every element of G by its contribution toward making G Abelian. Example 10.4.16. Let G be a group. Deﬁne a P(G)-fuzzy subset µ of G as follows: ∀x ∈ G, µ(x) = {z ∈ G | x ∈ z}. Then µ(e) = G. Let x, y ∈ G. Since x ∈ z if and only if x−1 ∈ z, µ(x) = µ(x−1 ). Since x, y ∈ z imply xy ∈ z, µ(xy) ⊇ µ(x) ∩ µ(y). Thus µ is a P(G)-fuzzy subgroup of G. If G is abelian, then µ measures x by the set of all elements of G of which x is a multiple. Example 10.4.17. Let R be a ring and let M1 and M2 be R-modules. Let X denote the set Hom(M1 , M2 ) of group homomorphisms of M1 into M2 . Then X is a group, where the operation is point-wise addition. Deﬁne a P(X)-fuzzy subset µ of X as follows: ∀f ∈ X, µ(f ) = {r ∈ R | rf (x) = f (rx) for all x ∈ M1 }. Then µ is a P(R)-fuzzy subgroup of X. Note that, µ()) = R, where )(x) = 0, ∀x ∈ M1 . We can think of µ as grading a group homomorphism from M1 into M2 by how close it comes to being an R-module homomorphism. Example 10.4.18. Let M be an R-module, where R is a ring. Let Aut(M ) be the group of all group automorphisms of M, where only the group structure of M is considered. For all f ∈ Aut(M ), let Mf = {r ∈ R | rf (x) = f (rx) for all x ∈ M }. Deﬁne the P(R)-fuzzy subset µ of Aut(M ) as follows: ∀f ∈Aut(M ), µ(f ) = Mf ∩ Mf −1 . Then µ is a P(R)-fuzzy subgroup of Aut(M ). Note that µ(i) = R, where i(x) = x ∀x ∈ M .

10.5 Applications The results in this section are mainly from [2]. Some of the results are also from [4]. The closure property of a fuzzy subgroup µ of a group (G, ·) can be obtained by the inequality µ(x · y) ≥ T (µ(x), µ(y)),where T is a t-norm. In Rosenfeld’s original deﬁnition, T was the function ‘minimum’. However, any t-norm provides a meaningful generalization of the closure property. We

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investigate two classes of fuzzy subgroups. The fuzzy subgroups in one class are called in [2] subgroup generated while those in the other are called function generated. Every fuzzy subgroup in these classes satisﬁes the above inequality with T given by T (a, b) = (a + b − 1) ∨ 0. It turns out that every fuzzy subgroup in either class is isomorphic to one in the other. We show that a fuzzy subgroup satisﬁes the above inequality with T = ∧ if and only if it is subgroup generated of a very special type. We then apply these notions to some abstract pattern recognition problems and to coding theory. Fuzzy subgroups of a group were ﬁrst deﬁned by Rosenfeld [18]. Subsequently his deﬁnition was generalized by Negoita and Ralescu [17] and by Anthony and Sherwood [1]. This section studies the structure of two classes of the more general fuzzy subgroups. That structure is used to characterize Rosenfeld’s original fuzzy subgroups. Whenever ∗ is introduced as a group operation it is usually suppressed and juxtaposition is used. Deﬁnition 10.5.1. A function T : [0, 1] × [0, 1] → [0, 1] is called a t-norm if the following properties hold for all x, y, z in [0, 1], (1) T (x, 1) = x, (2) T (x, y) ≤ T (z, y) if x ≤ z, (3) T (x, y) = T (y, x), (4) T (x, T (y, z)) = T (T (x, y), z). Some t-norms that are frequently encountered in the literature are Tm , P rod, and ∧, where Tm (x, y) = ∨{x + y − 1, 0} and P rod(x, y) = xy for all x, y ∈ G. Deﬁnition 10.5.2. Let (G, ∗) be a group. A function µ : G → [0, 1] is called a fuzzy subgroup of G with respect to a t-norm T if ∀ x, y ∈ G, (1) µ(x, y) ≥ T (µ(x), µ(y)); (2) µ(x−1 ) = µ(x); (3) µ(e) = 1, where e is the identity of G. A t-norm T1 is said to be stronger than a t-norm T2 if and only if T1 (x, y) ≥ T2 (x, y) for all x, y ∈ [0, 1]. Clearly, if µ is a fuzzy subgroup with respect to a t-norm T, then µ is a fuzzy subgroup with respect to any ‘weaker’ t-norm. Since ∧ is the strongest of all t-norms (see e.g. [19]), any fuzzy subgroup with respect to ∧ is a fuzzy subgroup with respect to any other t-norm. The following lemma provides some justiﬁcation for the inclusion of condition (3) in the deﬁnition of a fuzzy subgroup. Lemma 10.5.3. Let (G, ∗) be a group and µ a fuzzy subset of G such that the following conditions hold: (1) µ(xy) ≥ µ(x) ∧ µ(y), (2) µ(x−1 ) = µ(x),

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(3) µ(e) > 0. Deﬁne the function η of G into [0, 1] by for all x ∈ G, η(x) = µ (x) /µ (e) . Then η is a fuzzy subgroup of G with respect to ∧ such that η (e) = 1. Proof. For all x ∈ G, η(e) = µ(e)/µ(e) = µ(xx−1 )/µ(e) ≥ µ(x) ∧ µ(x−1 )/µ(e) = µ(x) ∧ µ(x)/µ(e) = µ(x)/µ(e) = η(x). Hence it follows that η (e) = 1 and that 0 ≤ η (x) ≤ 1 for all x ∈ G. Now η(xy) = µ(xy)/µ(e) ≥ µ(x) ∧ µ(y)/µ(e) = µ(x)/µ(e) ∧ µ(y)/µ(e) = η(x) ∧ η(y). Moreover, η(x−1 ) = µ(x−1 )/µ(e) = µ(x)/µ(e) = η(x). Therefore η is a fuzzy subgroup of G with respect to ∧. Deﬁnition 10.5.4. Let G1 and G2 be groups and let µ1 and µ2 be fuzzy subgroups of G1 and G2 , respectively, with respect to a t-norm T. The fuzzy subgroups µ1 and µ2 are called isomorphic if there an isomorphism f of G1 onto G2 such that µ1 = µ2 ◦ f. In the next result, we can think that the value of the fuzzy subgroup at a particular point x will be found in a randomly selected subgroup. This yields a particular way of generating fuzzy subgroups. We ﬁrst review some basic deﬁnitions. Let X be a set and let A be a collection of subsets of X. Then A is called an algebra of sets if (1) A∪B ∈ A whenever A, B ∈ A and (2) the complement of A, cA, is in A whenever A is in A. An algebra A of sets is called a σ-algebra if ∅ ∈ A and if every union of a countable collection of sets in A is again in A. A function P : A → R is called a probability measure if P (A) > 0∀A ∈ A\{∅}, P (Ω) = 1, and P (∪∞ i=1 Ai ) = ∞ P (Ai ) for any countable union of disjoint sets Ai , i = 1, 2, .... The triple Σi=1 (Ω, A, P ) is called a probability space. Theorem 10.5.5. Let (G, ∗) be a group and let S be the set of all subgroups of G. For all x ∈ G, let Sx = {S ∈ S | x ∈ S} and let S = {Sx | x ∈ G}. Let A be any σ-algebra on S which contains the σ-algebra generated by S and let m be a probability measure on (S, A) . Then µ : G → [0, 1] deﬁned by µ (x) = m(Sx ) for all x ∈ G is a fuzzy subgroup of G with respect to Tm . A fuzzy subgroup obtained in this manner is called subgroup generated. Proof. Let x, y ∈ G. Suppose S ∈ Sx ∩ Sy . Then S is a subgroup of G containing both x and y. Thus xy ∈ S. Hence S ∈ Sxy . Therefore, Sxy ⊇ Sx ∩ Sy . Now

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µ(xy) = m(Sxy ) ≥ m(Sx ∩ Sy ) = m(Sx ) + m(Sy ) − m(Sx ∪ Sy ) ≥ µ(x) + µ(y) − 1. Since µ (xy) ≥ 0, it follows that µ(xy) ≥ (µ(x) + µ(y) − 1) ∨ 0 = Tm (µ(x), µ(y)). Clearly, Sx−1 = Sx . Thus µ(x−1 ) = m(Sx−1 ) = m(Sx ) = µ(x). Moreover, Se = S. Hence µ(e) = µ(S) = 1. By Deﬁnition 10.5.2, µ is a fuzzy subgroup of G with respect to Tm . In the next result, we can think of a point which travels in some random fashion through a group and we compute the probability of ﬁnding the point in a particular subgroup. Theorem 10.5.6. Let (G, +) be a group and let H be a ﬁxed subgroup of G. Let (Ω, A, P ) be a probability space and (G, ⊕) be a group of functions mapping Ω into G with ⊕ deﬁned by point-wise addition in the range space. Assume that for all f ∈ G, Gf = {ω ∈ Ω | f (ω) ∈ H} is an element of A. Then ν : G → [0, 1] deﬁned by ν (f ) = P (Gt ) for all f ∈ G is a fuzzy subgroup of G with respect to Tm . A fuzzy subgroup obtained in this manner is called function generated. Proof. Let f, g ∈ G. Suppose ω ∈ Gf ∩ Gg . Then f (ω) ∈ H and g (ω) ∈ H. Since H is a subgroup of G, f (ω) + g (ω) = (f ⊕ g) (w) ∈ H. Thus w ∈ Gf ⊕g . Therefore, Gf ⊕g ⊃ Gf ∩ Gg . Now ν(f ⊕ g) = P (Gf ⊕g ) ≥ P (Gf ∩ Gg ) = P (Gf ) + P (Gg ) − P (Gf ∪ Gg ) ≥ ν(f ) + ν(g) − 1. Since ν(f ⊕ g) ≥ 0, it follows that ν(f ⊕ g) ≥ (ν(g) + ν(g) − 1) ∨ 0 = Tm (ν(f ), ν(g)). Note that Gf = Gf . Hence ν(*f ) = P (Gf ) = P (Gf ) = ν(f ). The identity in G is the function 0 : Ω → G deﬁned by 0(ω) = 0 for all ω ∈ Ω. Now G0 = Ω and so ν(0) = P (Ω) = 1. Therefore, ν is a fuzzy subgroup of G with respect to Tm . We next establish a basic equivalence between the notions of subgroup generated and function generated. Fuzzy subgroups with respect to the tnorm ∧ are then characterized in terms of these concepts. Theorem 10.5.7. Every function generated fuzzy subgroup is subgroup generated.

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Proof. Let ν be a function generated fuzzy subgroup with (G, +), H, (G, ⊕) and (Ω, A, P ) satisfying the properties given in Theorem 10.5.6. Let S be the family of all subgroups of G. For all ω ∈ Ω, let Sω = {f ∈ G | f (ω) ∈ H}. It follows easily that Sω is a subgroup of G for all ω ∈ Ω. Let σ : Ω → S be deﬁned by σ(ω) = Sω for all ω ∈ Ω. Let A and µ be the σ-algebra and measure induced on S by σ and the probability space (Ω, A,P ), that is, a collection A of subgroups of G is measurable if and only if σ −1 (A) = {ω ∈ Ω | Sω ∈ A} ∈ A and m(A) = P (σ −1 (A)). Consider subsets of S of the form Sf = {S ∈ S | f ∈ S}. Let Gf be the subset of Ω described in Theorem 10.5.6. It follows that σ −1 (Sf ) = Gf ∈ A. Hence Sf ∈ A and m(Sf ) = P (Gf ). Now (G, ⊕) and (S, A, m) satisfy the descriptions of (G, ·) and (S, A, m) given in Theorem 10.5.5. Deﬁne the fuzzy subset µ of G by µ(f ) = m(Sf ) for all f ∈ G. Then µ is a subgroup generated fuzzy subgroup of G by Theorem 10.5.5. However, µ(f ) = m(Sf ) = P (Gf ) = ν(f ) for every f ∈ G. Therefore, µ = ν and so ν is a subgroup generated. Theorem 10.5.8. Every subgroup generated fuzzy subgroup is isomorphic to a function generated fuzzy subgroup. Proof. Let µ be a subgroup generated fuzzy subgroup with (G, ·) and (S, A, m) satisfying the properties given in Theorem 10.5.5. For every subgroup S of G, let Gs = G. Let G = s∈S GS and H = S∈S S. It follows easily that G is a group with H a subgroup of G,where the operation + is inherited coordinatewise from the operation · in G. For all x ∈ G, let φx : S → G be deﬁned by φx (S) = ψx,S for all S ∈ S, where e if S ∗ = S, ψx,S (S ∗ ) = x if S ∗ = S. Let G = {φx | x ∈ G}. Deﬁne an operation ⊕ on G by point-wise addition in G, that is, (φx ⊕ φy )(S) = ψx,S + ψy,S = ψxy,S = φxy (S) for every S ∈ S. Now (G, ⊕) is a group and the function ξ : G → G deﬁned by ξ(x) = φ2 for every x ∈ G is an isomorphism. Consider the sets Gφx = {S ∈ S | φx (S) ∈ H}. Let Sx be the subset of S described in Theorem 10.5.5. Suppose S ∈ Gφx . Then φx (S) = ψx,S ∈ H. Hence ψx,S (S) = x ∈ S. Therefore, S ∈ Sx . Conversely, if S ∈ Sx , then x ∈ S and ψx,S = ψx,S (S) = x ∈ S. Hence S ∈ Sx . Therefore, Gφx = Sx ∈ A. Now (G, +), H, (S, A, m) and (G, ⊕) satisfy the descriptions of (G, +), H, (Ω, A, P ) and (G, ⊕) given in Theorem 10.5.6. Thus by Theorem 10.5.6, ν : G → [0, 1] deﬁned by ν (φx ) = m (Gφx ) for all φx ∈ G is a function generated fuzzy subgroup of G. Moreover, µ(x) = m(Sx ) = m(Gφx ) = ν(φx ) = ν ◦ ξ(x). Therefore, µ and ν isomorphic by Deﬁnition 10.5.4. The preceding two theorems show that the notions of function generated and subgroup generated are essentially equivalent. These ideas provide some basic intuition concerning the meaning of the values of certain fuzzy subgroups. An obvious problem is to decide which fuzzy subgroups are generated in this way. The next few results provide a partial solution to this problem

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which will include all the fuzzy subgroups originally introduced by Rosenfeld in [18]. First, we review some deﬁnitions. Let A be a set of real numbers and consider a countable collection {In } of open intervals which cover A, i.e., A ⊆ ∪n In . Let l(In ) denote the length of the interval In . Deﬁne the outer measure m∗ (A) of A to be the inﬁmum of all sums of the lengths of a cover, i.e., m∗ (A) = ∧{Σl(In )|A ⊆ ∪In }. A set E is said to be measurable if for every set A, m∗ (A) = m∗ (A ∩ E) + m∗ (A ∩ cA). Let E be a measurable set. Deﬁne the Lebesgue measure m(E) to be the outer measure of E. Then m is the set function obtained by restricting the set function m∗ to the family M of measurable sets. Theorem 10.5.9. Every fuzzy subgroup ν of G with respect to ∧ is subgroup generated. Proof. Let S be the collection of all subgroups of G and let η : [0, 1] → S be deﬁned by η(t) = νt for all t in [0, 1] . Let A and m be the σ-algebra and measure induced on S by the function η, using Lebesgue measure P, on [0, 1]. That is, a subset A of S is measurable if and only if η −1 (A) is Lebesgue measurable and then m(A) = P (η −1 (A)). Let x ∈ G and consider the set Sx = {S ∈ S | x ∈ S}. For every t in [0, ν (x)], ν(x) ≥ t and so / νt . Therefore, νt ∈ Sx if and only if x ∈ νt . Moreover, if t > ν(x), then x ∈ t ∈ [0, ν (x)]. Hence η −1 (Sx ) = [0, ν (x)] which is Lebesgue measurable. Now (G, ·) and (S, A, m) satisfy the conditions of Theorem 10.5.5. Hence µ : G → [0, 1] deﬁned by µ(x) = m(Sx ) for every x in G is a subgroup generated fuzzy subgroup. However, ν(x) = P [0, ν (x)] = P (η −1 (Sx )) = m(Sx ) = µ(x) for every x in G. Therefore, ν = µ and ν is subgroup generated. Theorem 10.5.10. Let µ be a subgroup generated fuzzy subgroup with (G, ·) and (S, A, m) and the sets Sx , for x ∈ G, as described in Theorem 10.5.5. If there exists S ∗ ∈ A which is linearly ordered by set inclusion such that m (S ∗ ) = 1, then µ is a fuzzy subgroup with respect to ∧. Proof. Let x, y ∈ G. Since S ∗ is linearly ordered, either Sx ∩ S ∗ ⊆ Sy ∩ S ∗ or Sy ∩ S ∗ ⊆ Sx ∩ S ∗ . We may assume without loss of generality that Sx ∩ S ∗ . Suppose that S ∈ Sx ∩ S ∗ . Then S ∈ Sy ∩ S ∗ . Hence both x and y are in S. Since S is a group, xy ∈ S and so S ∈ Sxy . Therefore, Sx ∩ S ∗ ⊆ Sxy . Now m(Sx ∩ S ∗ ) = m(Sx ). Also, m(Sxy ) ≥ m(Sx ∩ S ∗ ) = m(Sx ) ≥ m(Sx ) ∧ m(Sy ). Therefore, µ(xy) ≥ µ(x) ∧ µ(y) and so µ is a fuzzy subgroup with respect to ∧. Theorems 10.5.9 and 10.5.10 combine to yield the following characterization of fuzzy subgroups with respect to ∧. Theorem 10.5.11. A fuzzy subgroup is a fuzzy subgroup with respect to ∧ if and only if it is subgroup generated and the generating family possesses a subfamily of measure one which is linearly ordered by set inclusion.

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We now present an application for function generated fuzzy subgroups. We ﬁrst consider a generalized recognition problem. Suppose that F is a device which receives a stream of discrete inputs and produces a stream of discrete outputs. We make the following assumptions about F and about knowledge of the input and output. (1) F is deterministic and acts independently on each individual input. That is, a particular input produces the same output each time that it is provided to F. However, the output which is produced from any speciﬁc input is not known. (2) There is complete knowledge of the outputs. That is, the output stream is observable. (3) The input stream is not observable. The possible inputs are known and estimates can be obtained of their relative frequencies in a large segment of the input stream. (4) The outputs have an algebraic character in the sense that they can be identiﬁed with the objects in a group. Thus there is a method of combining the outputs which has the ordinary properties of a group operation. Let I denote the collection of inputs and let O denote the collection of outputs. If T ∈ I then F (T ) ∈ O. Thus F is identiﬁed with a function from I into O. Suppose that f is a known function of I into O. Moreover, suppose that some particular characteristic of F which we call “faithfullness” is associated with solvability for x of an equation in the output group of the form x + f (T ) = F (T ), where + is the group operation. If for some T ∈ I a solution for x can be found in a given subgroup H, then the output F (T ) is called H − f faithful to the input T. For a suﬃciently large ﬁnite segment of the output stream and for a given function f and subgroup H, We consider the problem of estimating the proportion of the outputs which are H − f faithful to their respective inputs. In order to translate this problem into the setting of fuzzy subgroups, certain identiﬁcations are necessary. The outputs have already been identiﬁed with a group (G, +). The inputs may be identiﬁed with a probability space (Ω, A, P ), where Ω = I, A is the power set of Ω, and P (T ) is the known estimate of the relative frequency of T in the input stream for each T ∈ Ω. If (G, ⊕) is the set of all functions from Ω into G with ⊕ deﬁned by point-wise addition in the range space, then both F and f may be identiﬁed with elements of G. The function f is known while F is not known. Also, H is a ﬁxed subgroup of G. By Theorem 10.5.6, the fuzzy subset ν of G deﬁned by ν(g) = P {T ∈ Ω | g(T ) ∈ H} is a function generated fuzzy subgroup of G with respect to Tm . Now ν(F ) can be estimated by observing throughput stream over some ﬁnite segment and computing the percentage of those outputs which are in H. Also, ν (*f ) = ν (f ) is a known quantity since the function f is known. An output, F (T ), is H −f faithful to T if and only if x+f (T ) = F (T ) has a solution for x which is in H. This occurs if and only if x = F (T ) − f (T ) = (F * f )(T ) ∈ H. Therefore, ν(F * f ) is the probability that F (T ) is H − f faithful to T. The solution to the original problem may now be identiﬁed with ν(F *f ). This may

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be estimated using ν(f ), an estimate of ν(F ), and the properties of the fuzzy subgroup, ν, in the following way: Since Tm (ν(F ), ν(*f )) = Tm (ν(F ), ν(f )) = (ν(F ) + ν(f ) − 1) ∨ 0 ≥ ν(F ) + ν(f ) − 1, we have ν(F * f ) ≥ Tm (ν(F ), ν(*f ))

(10.5.1)

= Tm (ν(F ), ν(f )) = (ν(F ) + ν(f ) − 1) ∨ 0 ≥ ν(F ) + ν(f ) − 1. Similarly, ν(F ) = ν(F * f + f ) ≥ Tm (ν(F * f ), ν(f ))

(10.5.2)

= (ν(F * f ) + ν(f ) − 1) ∨ 0 ≥ ν(F * f ) + ν(f ) − 1 and ν(f ) = ν(f * F ⊕ F ) ≥ Tm (ν(f * F ), ν(F )) = Tm (ν(F * f ), ν(F ))

(10.5.3)

= (ν(F * f ) + ν(F ) − 1) ∨ 0 ≥ ν(F * f ) + ν(F ) − 1 From (10.5.1) and (10.5.2), we obtain ν(F ) − (1 − ν(f )) ≤ ν(F * f ) ≤ ν(F ) + (1 − ν(F )).

(10.5.4)

From (10.5.1) and (10.5.3), we obtain ν(f ) − (1 − ν(F )) ≤ ν(F * f ) ≤ ν(f ) + (1 − ν(F )).

(10.5.5)

Thus we obtain the following estimate for the solution ν(F * f ) : |ν(F * f ) − ν(f ) ∧ ν(F )| ≤ 1 − ν(f ) ∨ ν(F ).

(10.5.6)

The estimate is close only when ν(f ) or ν(F ) is close to 1. However, if ν can be shown to be a fuzzy subgroup with respect to ∧ the situation changes considerably. Suppose ν is a fuzzy subgroup with respect to ∧. In this case (10.5.1), (10.5.2) and (10.5.3) become, respectively, ν(F * f ) ≥ ν(F ) ∧ ν(f ),

(10.5.1’)

10.5 Applications

291

ν(F ) ≥ ν(F * f ) ∧ ν(f ),

(10.5.2’)

ν(f ) ≥ ν(F * f ) ∧ ν(F ).

(10.5.3’)

Now if ν(f ) > ν(F ), then from (10.5.1’) we have that ν(F * f ) ≥ ν(F ) and from (10.5.2’) we have that ν(F ) ≥ ν(F * f ), i.e., ν(F * f ) = ν(F ). Similarly, if ν(f ) < ν(F ), then from (10.5.1’) we conclude that ν(F * f ) ≥ ν(f ) and from (10.5.3’) we conclude that ν(f ) ≥ ν(F * f ). Hence ν(F * f ) = ν(f ). Therefore, if ν(f ) = ν(F ), then ν(F * f ) = ν(f ) ∧ ν(F ) and so we know the solution exactly. Finally, if ν(F ) = ν(f ) the best one can say is ν(F ) = ν(f ) ≤ ν(F * f ) ≤ 1. We now consider a more speciﬁc recognition problem. Let n be a natural number. An n × n array of the integers 1, 2, 3, ..., n2 is called a pattern. Suppose that F is a machine which accepts input patterns and produces output patterns. Each pattern may be identiﬁed with a transformation in Sn2 , the permutation group of n2 objects, in the following way. k2 k1 kn+1 kn+2 ... ... kn2 −n kn2 −n+1

... kn 1 2 3 ... n2 ... k2n ⇔ ... ... k1 k2 k3 ... kn2 ... kn2

Thus F is identiﬁable with a function from Sn2 into Sn2 . An output pattern is called recognizable if it is a composition of translations and rotations of the input. There is a subgroup H of Sn2 such that an output pattern F (T ) is recognizable if and only if there exists a transformation T ∗ in H such that T ∗ ◦ T = F (T ). Suppose that estimates of the relative frequency of patterns in the input stream can be obtained. Let Ω = Sn2 , A be the power set of Sn2 and P be a probability measure on Sn2 obtained from the estimates of the relative frequency of input patterns. Now (Ω, A, P ) = (Sn2 , A, P ) and (G, +) = (Sn2 , ◦) with (G, ⊕) and H deﬁned appropriately. Let f1 : Sn2 → Sn2 be deﬁned by f1 (T ) = T for every T ∈ Sn2 . Then the output pattern F (T ) is recognizable if and only if the equation x ◦ f1 (T ) = F (T ) has a solution for x in H. This is the deﬁnition of F (T ) being H − f1 faithful to T. Note that ν(f1 ) = P {T ∈ Sn2 | f1 (T ) = T ∈ H} = P (H). From the discussion of the generalized recognition problem, the probability that the output is recognizable (H − f1 faithful) is ν(F * f1 ) which may be estimated using the inequality |ν(F * f1 ) − P (H) ∧ ν(F )| ≤ 1 = P (H) ∨ ν(F ).

292

10 Membership Functions From Similarity Relations

Once again, in the event ν can be shown to be a fuzzy subgroup with respect to ∧ we obtain ν(F * f ) = P (H) ∧ ν(F ) if P (H) = ν(F ); otherwise P (H) = ν(F ) ≤ ν(F * f ) ≤ 1. It should be remembered that P (H) is known and ν(F ) can be estimated by the percentage of outputs which are in H. We now consider the standard problem concerning the transmission of strings of 0’s and 1’s across a symmetric binary channel with noise. Let B = {0, 1} and B n denote the set of all binary n-tuples, n ≥ 2. Then B n is a group under componentwise addition modulo 2. Let C ⊆ B n denote the set of all codewords. Then C is a subgroup of B n . We make the following identiﬁcations: C = H = I = Ω and B n = G = O, where H, G, I, Ω, and O are as described above. In this situation, f is known, f (T ) is unknown, F is unknown, and F (T ) is known, where T ∈ H. We let f be the identity map since in the ideal situation their is no noise and so the output equals the input. We recall that f (T ) is observable and ν(F ) can be estimated. Since f is the identity map, ν(f ) = 1. Thus ν(F f ) = ν(F ) by equality (10.5.1). Thus F (T ) is H − f faithful. We now consider a more general situation. Assume that 1 ≥ ν(f ) > ν(F ). This is a reasonable assumption since f represents the ideal situation while F represents the real world situation. Then by inequality (10.5.1’), |ν(F f ) − ν(f ) ∧ ν(f )| ≤ 1 − ν(f ) ∨ ν(F ) and so |ν(F f ) − ν(F )| ≤ 1 − ν(f ). Now assume that ∀a, b ∈ Im(ν), a = b, 1 − ν(f ) < |a − b|. Then |ν(F f ) − ν(F )| = 0. Once again, F (T ) is H − f faithful. Consider the fuzzy coset νf , where νf (g) = ν(g f )∀g ∈ G. Then νf (F ) = ν(F ). Also, g ∈ (νf )a ⇔ νf (g) ≥ a ⇔ ν(g f ) ≥ a ⇔ g f ∈ νa ⇔ g ∈ f ⊕ νa . We now note a structure result for the group (G, ⊕) and the fuzzy subgroup ν of G. For all f ∈ G, (g ⊕ g)(T ) = g(T ) ⊕ g(T ) = 0 ∀T ∈ I. Thus 2G = {θ}, where θ(T ) = 0 ∀T ∈ I. Thus G = ⊕g∈G g. For all g ∈ G, deﬁne the fuzzy subset ν (g) of G as follows: ν (g) (g) = ν(g) if g ∈ ν ∗ and ν (g) (h) = 0 if h ∈ G\{θ, g}. Then ν (g) is a fuzzy subgroup of G. Hence ν = ⊕g∈S ν (g) for some subset S of ν ∗ , where |S| = |ν ∗ : Z2 | by [[12], Theorem 2.3, p. 96]. We now discuss some other applications of fuzzy group theory. The work in [11] is concerned with the classiﬁcation of knowledges when they are endowed with some algebraic structure. By using the quotient group of symmetric knowledges, an algebraic method is given in [11] to classify them. Also the anti-fuzzy subgroup construction is used to classify knowledges. In [8], fuzzy points are regarded as data and fuzzy objects are constructed from the set of given data on an arbitrary group. Using the method of least squares, optimal fuzzy subgroups are deﬁned for the set of data and it is shown that one of them is obtained as a fuzzy subgroup by a set of some modiﬁed data. In [20], a decomposition of an L-valued set (L a lattice) gives a family of characteristic functions which can be considered as a binary block-code.

References

293

Conditions are given under which an arbitrary block-code corresponds to an L-valued fuzzy set. An explicit description of the Hamming distance, as well as of any code distance, is also given, all in lattice-theoretic terms. A necessary and suﬃcient condition is given for a linear code to correspond to an L-valued fuzzy set. In such a case the lattice has to be Boolean.

References 1. J. M. Anthony and H. Sherwood, Fuzzy groups redeﬁned, J. Math. Anal. Appl. 69 (1979) 124-130. 284 2. J. M. Anthony and H. Sherwood, A characterization of fuzzy subgroups, Fuzzy Sets and Systems 7 (1982) 297-305. 283, 284 3. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, SpringerVerlag, New York, 1981 4. S-C Cheng and J. N. Mordeson, Applications of fuzzy algebra in automata theory and coding theory, Fifth IEEE International Conference on Fuzzy Systems, Proceedings vol. 1, 1996, 125-129. 283 5. P. S. Das, Fuzzy groups and level subgroups, J. Math Anal. Appl. 84 (1981) 264-269. 6. G. J. Klir, T.A. Folger, Fuzzy Sets, Uncertainty, and Information, 2nd ed., Prentice-Hall, Englewood Cliﬀs, NJ, 1994. 7. S. Kundu, Membership functions for a fuzzy group from similarity relations, Fuzzy Sets and Systems 101 (1999) 391-402. 267 8. T. Kuraoka and N-Y Suzuki, Optimal fuzzy objects for the set of given data in the case of the group theory, Inform. Sci. 92 (1996) 197-210. 292 9. R. Lowen, Convex fuzzy sets, Fuzzy Sets and Systems 19 (1980) 291-310. 10. D. S. Malik, J, N. Mordeson, and P. S. Nair, Fuzzy generators and fuzzy direct sums of abelian groups, Fuzzy Sets and Systems 50 (1992) 193-199. 11. M. Mashinchi and M. Mukaidono, Algebraic knowledge classiﬁcation, J. Fuzzy Math. 2 (1994) 233-247. 292 12. J. N. Mordeson, Fuzzy subﬁelds of ﬁnite ﬁelds, Fuzzy Sets and Systems 52 (1992) 93-96. 292 13. J. N. Mordeson, Bases of fuzzy vector spaces, Inf. Sci. 67 (1993) 87-92. 14. J. N. Mordeson, Bases of fuzzy algebraic substructures, Fuzzy Sets and Systems 62 (1994) 185-191. 15. G. C. Muganda and M. Garzon, On the structure of fuzzy groups, In P. P. Wang, editor, Advances in Fuzzy Theory and Technology, Vol. I, Book Wrights, Durham North Carolina, 1993, 23-42. 16. G. C. Muganda, Fuzzy algebras based on algebraic properties, J. Fuzzy Math. 6 (1998) 649-658. 280 17. C. V. Negoita and D. A. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, 1975, 54-59. 284 18. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. 284, 288 19. B. Schweizer and A. Sklar, Statistical metric spaces, Paciﬁc J. Math. 10 (1960) 313-334. 284 20. B. Seselja, A. Tepavcevic, and G. Vojvodic, L-fuzzy sets and codes, Fuzzy Sets and Systems 53 (1993) 217-222. 292

294

10 Membership Functions From Similarity Relations

21. F-G Shi, L-fuzzy relations and L-fuzzy subgroups, J. Fuzzy Math. 8(2000) 491499. 22. Z. Wang, Y. Yu and F. Dai, On T-congruence L-relations on groups and rings, Fuzzy Sets and Systems 119 (2001) 393-407.

Index

a-cut 2 a-level set 2 Abelian fuzzy subset 9 admissible 73 algebra 285 algebra of sets 285 ascending central series associated 191 basic fuzzy subgroup

conormal pair 82 contains 2 core 112 coset 12 left 12 right 12 crisp power set 239 68

158

central chain 79 central series 80 centralizer 61 characteristic function 2 class 68 closed under projections 244 closure torsion 147 closure of fuzzy subgroup 112 commutative 61 commutator 42, 73 subgroup 73 complete direct product 3, 20, 21 complete Heyting algebra 280 components 209, 226 composition chain 94 composition length 171 conjugacy classes 42, 44 conjugate fuzzy subgroup 10 conjugate fuzzy subgroups 46 conjugate to 46 conjugates 42

Dedekind 91 Dedekind Theorem 95 derived chain 84 derived chain of a subgroup 84 derived series 85 descending central chain 73, 79 descending central series 80 direct product 22, 26 complete 20, 21 weak 22, 26 direct sum of fuzzy subgroups 193 distinct 205, 226 divisible fuzzy subgroup 147 epimorphism 127 equilibrium 131 equivalent 167, 169 Extension Principle 4 f -invariant 20 factor fuzzy subgroup factor group 13 faithful 289 ﬁnite n-chain 209 ﬁnite fuzzy subset 1 ﬂag 226

14, 17

296

Index

components of 226 free (s, t]-fuzzy subgroup 136 free fuzzy basis 127 free fuzzy subgroup 121 free group 120 fully invariant 99 function generated 286 fuzzy p-subgroup 177 subsemigroup 23 abelian 55 characteristic subgroup 45 cyclic p-subgroup 188 cyclic subgroup 187 point 2 singleton 2 submonoid 23 torsion 193 fuzzy p-subgroup 177 fuzzy p∗ -subgroup 195 fuzzy pi -primary components 191 fuzzy Abelian 55 fuzzy C subset 245 fuzzy complement 131 equilibrium of 131 fuzzy coset left 156 fuzzy cyclic p-subgroup 188 fuzzy direct product 168 fuzzy function 127 fuzzy isomorphic fuzzy subgroups 207 fuzzy Lagrange’s Theorem 56 fuzzy order 32, 58 fuzzy point 2 fuzzy power set 1 fuzzy quotient group 44, 57 fuzzy set associated to 191 fuzzy singleton 2, 145 foot 120 level 120 fuzzy solvable 57 fuzzy subgroup 6 f -invariant 20 p-primary 146 basic 158 central chain of 79 characteristic 73, 99 closure of 112

conjugate 10 core 112 cyclic 158 derived chain 84 derived series 85 direct sum 193 divisible 147 factor 14, 17 fully invariant 73, 99 fuzzy solvable 57 generalized characteristic 99 generated by 8 homomorphic image of 121 length of 171 normal 73 principal length 171 normal 10, 15 pure 150 quasinormal 105, 107 quotient 14, 17 reduced 149 solvable 85 solvable series 86 solvable series for 86 subnormal 110 composition length 171 torsion 146 fuzzy subgroupoid 41 fuzzy subgroups central chain of 79 descending central chain of 79 descending central series 80 distinct 205 fuzzy pi -primary components 191 fuzzy submonoid 23 fuzzy subsemigroup 23 fuzzy subset 1 Abelian 9 admissible centralizer of 61 class of 68 complete direct product of 3 contained in 2 contains 2 ﬁnite 1 image of 1 inﬁnite 1 intersection of 2 inverse of 6

Index normal 61 normalizer of 61 product of 6 properly contained 2 properly contains 2 right invariant 268 support of a 1 tip of 61 translation invariant 268 union of 2 fuzzy subsets closed under projections 244 strongly equivalent 203 fuzzy torsion 193 fuzzy weak direct product 175 fuzzy central series of 80 generalized characteristic fuzzy subgroup 99 generate 8 generating levels 121 generating set 140 minimal 140, 154 generators 121 group deﬁning generators of 121 deﬁning relations 121 presentation of 121 group factor 13 quotient 13 groupoid 41 Hall 93 Hamiltonian 91 height of 121 homomorphic 17 homomorphism 17 weak 17 identity property 277 image 4 independent generators 144 index 45, 54 inﬁmum 1 inﬁnite fuzzy order 32, 58 inﬁnite fuzzy subset 1 interlocking position 209

intersection of fuzzy subsets 2 invariants of fuzzy subgroups 154 inverse image 4 inverse of a fuzzy subset 6 isomorphic 17, 235, 285 isomorphism 17 weak 17 k-pad keychain 209 index of 209 padidity 209 keychain 209, 226 keychains distinct 226 L-fuzzy subset 280 Lebesgue measure 288 left coset 12 left fuzzy coset 156 left ideal 109 left-invariance 268 length 128, 171 level of 121 level subgroups 94 linearly dependent 145 linearly independent 145 maximal 145 maximally p-independent 161 Metatheorem 244 minimal generating set 140, 154 monoid 127 nilpotent 68, 73, 80 of class 80 nilpotent of class 73 normal 61 normal fuzzy subgroup normalizer 61 normalizer 11 operator domain 73 order 55 outer measure 288 (p,q)-group 97 p-basic 160 p-basis 160 P -group 95 P -Hall subgroup

95

10, 15

297

298

Index

p-independent 160 maximally 161 p-primary component 146 fuzzy subgroup 146 p-pure 159 padidity 209 pattern 291 periodic 95 pinned-ﬂag 226 pins 209 power set crisp 239 pre-image 4 presentation 121, 125 principal length 171 probability measure 285 probability space 285 product of fuzzy subsets 6 properly contains 2 property ∗ 183 pure fuzzy subgroup 150 quasinormal 104, 105 quotient fuzzy subgroup quotient group 13 recognizable 291 reduced form 120 reduced fuzzy subgroup relators 125 right coset 12 right ideal 109 right invariant 268

14, 17

149

(s, t]-fuzzy subgroup 132 (s, t]-fuzzy subgroup free 136 (s,t]-fuzzy subgroup generated by 135 (s, t]-homomorphic image 135 S ∗ -equivalent 235 semisimple 109 similarity relation 268 simple 94 SL(2, 3) group 105 solvable series 86

solvable series of a fuzzy subgroup 86 solvable subgroup 84 strong level subgroup 249 strongly equivalent fuzzy subsets 203 Subdirect Product Theorem 243 subgroup index of 55 nilpotent 73 subgroup generated 285 subnormal 105, 110, 167 subnormal series 105 subwords 120 sup property 1 support of fuzzy subset 1 supremum 1 surjective 127 Sylow p-subgroup 95 symmetrical alphabet 120 t-fuzzy relation 248 reﬂexive 248 equivalence 248 t-norm 284 stronger 284 tip 61 tip of fuzzy subgroup 81 torsion 139 torsion closure 147 fuzzy subgroup 146 torsion-free 139 translation invariant 268 union of fuzzy subsets

2

weak direct product 22, 26 homomorphism 17 isomorphism 17 product 21, 22 weakly homomorphic 17 isomomorphic 17 word index set of 121 inverse of 121 words 120

Index of Symbols

aY 2 (A)(r) 257 [a,b] 73 C 256 C(X × X) 256 CI(G × G) 257 CI(X × X) 256 CC 257 CC 256 J 257 CC Ct (G) 248 E(G) 248 Et (G) 248 FP(X) fuzzy power set 1 FP(X × X) 256 f (µ) 4 f −1 (ν) 4 F (X) free group over X 120 F Oµ (x) 32 [G : µ] index of 45 GCFS 99 G/µ 12 H-fuzzy subgroup 280 I nonempty index set 1 I(µ) or µ(X) image of µ 1 LI(X) 255 LJC 255 257 Lnt Lnt (G) 248 Lnst 257 Ln (G) 248 Lt (G) 248 (m, n) 31

N set of positive integers 1 N F (G) 10 N(A(r)) 257 N G 41 O(µ) 58 o(µ) 55 o(x) 31 (p,q)-group 105 Q set of rational numbers 1 R set of real numbers 1 R Representation Function 240 S system of fuzzy singletons 145 S ∗ 145 Sa 145 T ∗ ◦ T = F (T ) 291 ya or ay fuzzy point 2 Z set of integers 1 [λ, µ] 73 (λ, µ) 73 [µ : ν], index of ν in µ 56 Ω 291 ∩i∈I µi 2 ∩ 2 ∪i∈I µi 2 ∪ 2 ∅ emptyset 1 η 135

η 135

µ 248

µ 8 µ◦ν 6 µ ν 15 2 3 µ1 ⊗µ

300

Index of Symbols

µ∗ support of µ 1 µ1 8 µG 112 µ(p) 146 µ−1 6 µθ 45 µg 46 µn 8 µa α-cut or α-level set µN quotient of µ by N µ∗ 6 ν (g) 292

≺ η Xi i∈I ∼ i∈I

2 124

µi

133 3 3

⊂ properly contained in 2 ⊆ contained in 2 ⊃ properly contains 2 ⊇ contains 2 ∨i∈I µi 2 ∨ maximum or supremum 1 ∧i∈I µi 2 ∧ minimum or inﬁmum 1